| L(s) = 1 | − 3·2-s + 4-s + 10·5-s + 14·7-s − 3·8-s − 30·10-s − 62·11-s − 6·13-s − 42·14-s + 9·16-s − 40·17-s − 122·19-s + 10·20-s + 186·22-s − 16·23-s + 75·25-s + 18·26-s + 14·28-s − 352·29-s + 66·31-s + 165·32-s + 120·34-s + 140·35-s − 188·37-s + 366·38-s − 30·40-s − 16·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s + 0.894·5-s + 0.755·7-s − 0.132·8-s − 0.948·10-s − 1.69·11-s − 0.128·13-s − 0.801·14-s + 9/64·16-s − 0.570·17-s − 1.47·19-s + 0.111·20-s + 1.80·22-s − 0.145·23-s + 3/5·25-s + 0.135·26-s + 0.0944·28-s − 2.25·29-s + 0.382·31-s + 0.911·32-s + 0.605·34-s + 0.676·35-s − 0.835·37-s + 1.56·38-s − 0.118·40-s − 0.0609·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 T + p^{3} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 62 T + 3582 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 3378 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 40 T + 8750 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 122 T + 17398 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 16 T + 34 p T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 352 T + 78278 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 66 T + 45870 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 188 T + 44542 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 18350 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 396 T + 2226 p T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 p T + 15582 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 982 T + 504354 T^{2} + 982 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 516 T + 418118 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 880 T + 575238 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 356 T + 100374 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 310 T + 664038 T^{2} + 310 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 326 T + 789802 T^{2} - 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 680 T + 744870 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 796 T + 411158 T^{2} + 796 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 670 T + 1145410 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.73367796985506401003902973602, −10.71628834787991071714588334763, −9.915843558118206884024008710122, −9.589166291319292280483701314920, −9.088519870035481934270409011195, −8.749022061685460881585497054662, −8.112804230021044235271879989329, −7.919320767137666454790690186965, −7.34645259867073888196785873983, −6.57278425894894481640026427981, −6.11845432029063796772736407004, −5.57744502018219020086240826481, −4.79606338446240584283964420960, −4.71277872264457071646378309301, −3.57332554106405253211400630797, −2.72285383104770266038427571937, −2.06831059306442767213738978273, −1.56122235597130864247503891068, 0, 0,
1.56122235597130864247503891068, 2.06831059306442767213738978273, 2.72285383104770266038427571937, 3.57332554106405253211400630797, 4.71277872264457071646378309301, 4.79606338446240584283964420960, 5.57744502018219020086240826481, 6.11845432029063796772736407004, 6.57278425894894481640026427981, 7.34645259867073888196785873983, 7.919320767137666454790690186965, 8.112804230021044235271879989329, 8.749022061685460881585497054662, 9.088519870035481934270409011195, 9.589166291319292280483701314920, 9.915843558118206884024008710122, 10.71628834787991071714588334763, 10.73367796985506401003902973602
