| L(s) = 1 | + 8·2-s + 12·3-s − 210·5-s + 96·6-s − 512·8-s + 2.18e3·9-s − 1.68e3·10-s − 1.09e3·11-s − 2.76e3·13-s − 2.52e3·15-s − 4.09e3·16-s + 1.47e4·17-s + 1.74e4·18-s − 3.99e4·19-s − 8.73e3·22-s − 6.87e4·23-s − 6.14e3·24-s + 7.81e4·25-s − 2.21e4·26-s + 7.70e4·27-s − 2.05e5·29-s − 2.01e4·30-s + 2.27e5·31-s − 1.31e4·33-s + 1.17e5·34-s − 1.60e5·37-s − 3.19e5·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.256·3-s − 0.751·5-s + 0.181·6-s − 0.353·8-s + 9-s − 0.531·10-s − 0.247·11-s − 0.348·13-s − 0.192·15-s − 1/4·16-s + 0.725·17-s + 0.707·18-s − 1.33·19-s − 0.174·22-s − 1.17·23-s − 0.0907·24-s + 25-s − 0.246·26-s + 0.752·27-s − 1.56·29-s − 0.136·30-s + 1.37·31-s − 0.0634·33-s + 0.513·34-s − 0.521·37-s − 0.944·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.026229764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026229764\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 p T - 227 p^{2} T^{2} - 4 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 42 p T - 1361 p^{2} T^{2} + 42 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 1092 T - 18294707 T^{2} + 1092 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 1382 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14706 T - 194072237 T^{2} - 14706 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 39940 T + 701331861 T^{2} + 39940 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 68712 T + 1316513497 T^{2} + 68712 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 102570 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 227552 T + 24267298593 T^{2} - 227552 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 160526 T - 69163280457 T^{2} + 160526 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10842 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 630748 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 472656 T - 283219426127 T^{2} - 472656 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1494018 T + 1057378644487 T^{2} - 1494018 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2640660 T + 4484433750781 T^{2} - 2640660 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 827702 T - 2457652235217 T^{2} - 827702 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 126004 T - 6044834597307 T^{2} - 126004 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1414728 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 980282 T - 10086445719573 T^{2} - 980282 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3566800 T - 6481846746159 T^{2} - 3566800 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 5672892 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 11951190 T + 98599607520571 T^{2} + 11951190 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8682146 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.63031718916774618443500625102, −12.55516311602722165105178996562, −11.76971977034836977793949136947, −11.46150716690675842475739605697, −10.57292256125683498915466844564, −10.03146377584521989716027422144, −9.832865226955374622410573937025, −8.667945695100362402960433352417, −8.456696712534841751743613664089, −7.78194782654030496789757840601, −7.02211500773927800202802300755, −6.71152000501577677689600751908, −5.72488250731941276301291371887, −5.12795365530475347596602945477, −4.26255192786789488338566611610, −4.04583766788921321359098881765, −3.21738116624079064952079907792, −2.36027130641956334447531448500, −1.48030893478338029826240809088, −0.39101113651558714819590160300,
0.39101113651558714819590160300, 1.48030893478338029826240809088, 2.36027130641956334447531448500, 3.21738116624079064952079907792, 4.04583766788921321359098881765, 4.26255192786789488338566611610, 5.12795365530475347596602945477, 5.72488250731941276301291371887, 6.71152000501577677689600751908, 7.02211500773927800202802300755, 7.78194782654030496789757840601, 8.456696712534841751743613664089, 8.667945695100362402960433352417, 9.832865226955374622410573937025, 10.03146377584521989716027422144, 10.57292256125683498915466844564, 11.46150716690675842475739605697, 11.76971977034836977793949136947, 12.55516311602722165105178996562, 12.63031718916774618443500625102
