| L(s) = 1 | + 2·2-s + 7·3-s + 19·5-s + 14·6-s − 14·7-s + 27·9-s + 38·10-s + 72·13-s − 28·14-s + 133·15-s + 46·17-s + 54·18-s + 20·19-s − 98·21-s − 428·23-s + 125·25-s + 144·26-s − 120·29-s + 266·30-s − 117·31-s − 32·32-s + 92·34-s − 266·35-s + 201·37-s + 40·38-s + 504·39-s + 228·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s + 1.69·5-s + 0.952·6-s − 0.755·7-s + 9-s + 1.20·10-s + 1.53·13-s − 0.534·14-s + 2.28·15-s + 0.656·17-s + 0.707·18-s + 0.241·19-s − 1.01·21-s − 3.88·23-s + 25-s + 1.08·26-s − 0.768·29-s + 1.61·30-s − 0.677·31-s − 0.176·32-s + 0.464·34-s − 1.28·35-s + 0.893·37-s + 0.170·38-s + 2.06·39-s + 0.868·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.260797320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.260797320\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_4$ | \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 - 7 T + 22 T^{2} + 35 T^{3} - 839 T^{4} + 35 p^{3} T^{5} + 22 p^{6} T^{6} - 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 19 T + 236 T^{2} - 2109 T^{3} + 10571 T^{4} - 2109 p^{3} T^{5} + 236 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 2 p T - 3 p^{2} T^{2} - 20 p^{3} T^{3} - 19 p^{4} T^{4} - 20 p^{6} T^{5} - 3 p^{8} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 72 T + 2987 T^{2} - 56880 T^{3} - 2467079 T^{4} - 56880 p^{3} T^{5} + 2987 p^{6} T^{6} - 72 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 46 T - 2797 T^{2} + 354660 T^{3} - 2572699 T^{4} + 354660 p^{3} T^{5} - 2797 p^{6} T^{6} - 46 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 20 T - 6459 T^{2} + 266360 T^{3} + 38975081 T^{4} + 266360 p^{3} T^{5} - 6459 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 107 T + p^{3} T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 + 120 T - 9989 T^{2} - 4125360 T^{3} - 251421479 T^{4} - 4125360 p^{3} T^{5} - 9989 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 117 T - 16102 T^{2} - 5369481 T^{3} - 148534595 T^{4} - 5369481 p^{3} T^{5} - 16102 p^{6} T^{6} + 117 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 201 T - 10252 T^{2} + 12241905 T^{3} - 1941328349 T^{4} + 12241905 p^{3} T^{5} - 10252 p^{6} T^{6} - 201 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 228 T - 16937 T^{2} + 19575624 T^{3} - 3295927295 T^{4} + 19575624 p^{3} T^{5} - 16937 p^{6} T^{6} - 228 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 242 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - 96 T - 94607 T^{2} + 19049280 T^{3} + 7993651681 T^{4} + 19049280 p^{3} T^{5} - 94607 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 458 T + 60887 T^{2} - 40299420 T^{3} - 27521808259 T^{4} - 40299420 p^{3} T^{5} + 60887 p^{6} T^{6} + 458 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 435 T - 16154 T^{2} - 96366855 T^{3} - 38601889559 T^{4} - 96366855 p^{3} T^{5} - 16154 p^{6} T^{6} + 435 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - 668 T + 219243 T^{2} + 5168984 T^{3} - 53216876695 T^{4} + 5168984 p^{3} T^{5} + 219243 p^{6} T^{6} - 668 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 439 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - 1113 T + 880858 T^{2} - 582040011 T^{3} + 332541764605 T^{4} - 582040011 p^{3} T^{5} + 880858 p^{6} T^{6} - 1113 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 72 T - 383833 T^{2} + 55645200 T^{3} + 145311107761 T^{4} + 55645200 p^{3} T^{5} - 383833 p^{6} T^{6} - 72 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 70 T - 488139 T^{2} + 68682460 T^{3} + 235863792221 T^{4} + 68682460 p^{3} T^{5} - 488139 p^{6} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 358 T - 443623 T^{2} - 363516780 T^{3} + 123518857061 T^{4} - 363516780 p^{3} T^{5} - 443623 p^{6} T^{6} + 358 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 895 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 + 409 T - 745392 T^{2} - 678148585 T^{3} + 402936381551 T^{4} - 678148585 p^{3} T^{5} - 745392 p^{6} T^{6} + 409 p^{9} T^{7} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.257766015591938386314559981685, −8.195745463055743674995632602292, −8.157713138005833121128117569675, −7.54811730588280460829239510300, −7.44620541960707111578933655687, −6.96763576051939353725265584257, −6.52664048551735816706365108341, −6.37493802621862305781316296310, −6.23549125188950712254349521776, −5.97027251957612649653787012148, −5.54757376074780306613960118372, −5.52980651182890151205015686083, −4.94434783191715792275016143541, −4.91522148248426070731153771274, −4.06478753854614784121326647162, −3.87667382783758387352142483242, −3.78545497740369775409294383016, −3.54861781281908589959356630401, −3.14307598092340227930239193926, −2.61317853906157991140279702254, −2.13455761894216398315631848491, −1.90343578168075336024185046231, −1.83966700925145580868575068189, −1.10125095059929577839180841503, −0.39167957612536145646530025851,
0.39167957612536145646530025851, 1.10125095059929577839180841503, 1.83966700925145580868575068189, 1.90343578168075336024185046231, 2.13455761894216398315631848491, 2.61317853906157991140279702254, 3.14307598092340227930239193926, 3.54861781281908589959356630401, 3.78545497740369775409294383016, 3.87667382783758387352142483242, 4.06478753854614784121326647162, 4.91522148248426070731153771274, 4.94434783191715792275016143541, 5.52980651182890151205015686083, 5.54757376074780306613960118372, 5.97027251957612649653787012148, 6.23549125188950712254349521776, 6.37493802621862305781316296310, 6.52664048551735816706365108341, 6.96763576051939353725265584257, 7.44620541960707111578933655687, 7.54811730588280460829239510300, 8.157713138005833121128117569675, 8.195745463055743674995632602292, 8.257766015591938386314559981685
