| L(s) = 1 | + (−1.64 + 2.85i)2-s + (22.5 + 38.9i)3-s + (250. + 433. i)4-s + (−142. − 246. i)5-s − 148.·6-s + (−5.19e3 − 8.99e3i)7-s − 3.33e3·8-s + (8.82e3 − 1.52e4i)9-s + 939.·10-s − 8.21e3·11-s + (−1.12e4 + 1.95e4i)12-s + (2.32e4 + 4.03e4i)13-s + 3.42e4·14-s + (6.41e3 − 1.11e4i)15-s + (−1.22e5 + 2.12e5i)16-s + (2.97e5 − 5.14e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0728 + 0.126i)2-s + (0.160 + 0.277i)3-s + (0.489 + 0.847i)4-s + (−0.101 − 0.176i)5-s − 0.0467·6-s + (−0.817 − 1.41i)7-s − 0.288·8-s + (0.448 − 0.776i)9-s + 0.0296·10-s − 0.169·11-s + (−0.156 + 0.271i)12-s + (0.226 + 0.391i)13-s + 0.238·14-s + (0.0326 − 0.0566i)15-s + (−0.468 + 0.811i)16-s + (0.862 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.49789 - 0.732806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49789 - 0.732806i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-5.79e6 + 9.81e6i)T \) |
| good | 2 | \( 1 + (1.64 - 2.85i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-22.5 - 38.9i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (142. + 246. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (5.19e3 + 8.99e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + 8.21e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-2.32e4 - 4.03e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-2.97e5 + 5.14e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (6.98e4 + 1.21e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 - 1.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.27e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.68e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (9.87e6 + 1.71e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + 1.02e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (2.55e7 - 4.42e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (1.54e7 - 2.67e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (7.19e7 + 1.24e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.07e8 - 1.86e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-6.76e7 - 1.17e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 1.59e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.96e8 + 3.40e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.28e8 + 3.95e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-2.54e8 + 4.41e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 5.29e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.11532513538558991336115746339, −12.95249014280667200143900696606, −11.87382154527345756010812153040, −10.38773978020608147130315016078, −9.175208718991932402569816375175, −7.51064989840504884194398803622, −6.65111859509046706071417718510, −4.20542863503344914900040020846, −3.10952132880280033826074830588, −0.62807495444733271736667524393,
1.57210345124137976239302336640, 2.91938708916665969515528179691, 5.42153211118007475923468580086, 6.48248584703387259822008292796, 8.182355666082684695224210368489, 9.703589949375303287639201143787, 10.74451922521549862569344090415, 12.17612812389865441235420928772, 13.20485300340211999537195638301, 14.83683071493380443046222242491
