| L(s) = 1 | + 4·2-s + 12·4-s + 10·5-s + 32·8-s + 40·10-s − 54·11-s − 26·13-s + 80·16-s − 68·17-s − 42·19-s + 120·20-s − 216·22-s − 294·23-s + 75·25-s − 104·26-s − 312·29-s + 170·31-s + 192·32-s − 272·34-s − 408·37-s − 168·38-s + 320·40-s − 140·41-s − 322·43-s − 648·44-s − 1.17e3·46-s + 184·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s − 1.48·11-s − 0.554·13-s + 5/4·16-s − 0.970·17-s − 0.507·19-s + 1.34·20-s − 2.09·22-s − 2.66·23-s + 3/5·25-s − 0.784·26-s − 1.99·29-s + 0.984·31-s + 1.06·32-s − 1.37·34-s − 1.81·37-s − 0.717·38-s + 1.26·40-s − 0.533·41-s − 1.14·43-s − 2.22·44-s − 3.76·46-s + 0.571·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 366 T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 54 T + 3266 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 p T + 9702 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 42 T - 3246 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 294 T + 45538 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 312 T + 72134 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 170 T + 66762 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 408 T + 137142 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 140 T + 113862 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 322 T + 183130 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 184 T + 157790 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 116 T + 22638 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 382 T + 304434 T^{2} - 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 48 T + 389558 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1620 T + 1220646 T^{2} + 1620 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 930 T + 910922 T^{2} + 930 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 609518 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 532 T + 950254 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 120 T - 163546 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 292 T + 1261974 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1092 T + 2030982 T^{2} + 1092 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.214694724747001913503656623044, −8.783313211056937331586899807075, −8.184369870862272900661559043450, −7.982291473773302994582390475275, −7.25714343214328360981141888914, −7.19963177845694232817582806339, −6.31588748315711336547837759631, −6.29788695028113036436933709509, −5.62583862742707416781816961532, −5.48357618986033950302677894549, −4.75369992168316906850129724060, −4.69586675967311775602549626958, −3.90773492046865477091655470165, −3.55432068542527746499324599312, −2.84048271103812960000059241008, −2.36733579974574379302706016527, −1.91931416457233203967375547040, −1.64440331943189202411754721076, 0, 0,
1.64440331943189202411754721076, 1.91931416457233203967375547040, 2.36733579974574379302706016527, 2.84048271103812960000059241008, 3.55432068542527746499324599312, 3.90773492046865477091655470165, 4.69586675967311775602549626958, 4.75369992168316906850129724060, 5.48357618986033950302677894549, 5.62583862742707416781816961532, 6.29788695028113036436933709509, 6.31588748315711336547837759631, 7.19963177845694232817582806339, 7.25714343214328360981141888914, 7.982291473773302994582390475275, 8.184369870862272900661559043450, 8.783313211056937331586899807075, 9.214694724747001913503656623044
