| L(s) = 1 | + 2·2-s − 11·4-s + 10·5-s − 16·7-s − 36·8-s + 20·10-s + 26·11-s − 26·13-s − 32·14-s + 61·16-s − 60·17-s − 220·19-s − 110·20-s + 52·22-s − 52·23-s − 143·25-s − 52·26-s + 176·28-s − 58·29-s − 294·31-s + 358·32-s − 120·34-s − 160·35-s + 312·37-s − 440·38-s − 360·40-s − 40·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.37·4-s + 0.894·5-s − 0.863·7-s − 1.59·8-s + 0.632·10-s + 0.712·11-s − 0.554·13-s − 0.610·14-s + 0.953·16-s − 0.856·17-s − 2.65·19-s − 1.22·20-s + 0.503·22-s − 0.471·23-s − 1.14·25-s − 0.392·26-s + 1.18·28-s − 0.371·29-s − 1.70·31-s + 1.97·32-s − 0.605·34-s − 0.772·35-s + 1.38·37-s − 1.87·38-s − 1.42·40-s − 0.152·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68121 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 15 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 243 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 550 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 26 T + 93 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 p T + 19 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 60 T + 10078 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 220 T + 23770 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 52 T + 20402 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 294 T + 2363 p T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 312 T + 119370 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 40 T + 100154 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 322 T + 126453 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 130 T + 126173 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1002 T + 518987 T^{2} + 1002 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 900 T + 490250 T^{2} - 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 948 T + 615270 T^{2} + 948 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 320 T + 158614 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 660 T + 822410 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 648 T + 63810 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 258 T + 770157 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1212 T + 1502618 T^{2} + 1212 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 760 T + 1009370 T^{2} + 760 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 1157322 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.10525879163492887527541723861, −11.05330249343104460068820473000, −10.02755383433594352635505502822, −9.836793271769525853662912436370, −9.308553087093702942066863163329, −9.201950357206464568879163965787, −8.311006358282668005172840746282, −8.252881312175006003290966945010, −7.16934689259892465716986253978, −6.51850845579849506620335615069, −5.96745070307869472414828119554, −5.94208607785189985064744792427, −4.86688972439925716027236394618, −4.58630612414042137200423145805, −3.84208840596687593787346491843, −3.56636673840461775327808862451, −2.39733452292507130240137288298, −1.78152601391236604656149982717, 0, 0,
1.78152601391236604656149982717, 2.39733452292507130240137288298, 3.56636673840461775327808862451, 3.84208840596687593787346491843, 4.58630612414042137200423145805, 4.86688972439925716027236394618, 5.94208607785189985064744792427, 5.96745070307869472414828119554, 6.51850845579849506620335615069, 7.16934689259892465716986253978, 8.252881312175006003290966945010, 8.311006358282668005172840746282, 9.201950357206464568879163965787, 9.308553087093702942066863163329, 9.836793271769525853662912436370, 10.02755383433594352635505502822, 11.05330249343104460068820473000, 11.10525879163492887527541723861
