| L(s) = 1 | + 2-s + 3-s + 4-s + 0.380·5-s + 6-s + 1.68·7-s + 8-s + 9-s + 0.380·10-s − 3.77·11-s + 12-s − 5.19·13-s + 1.68·14-s + 0.380·15-s + 16-s − 17-s + 18-s + 1.35·19-s + 0.380·20-s + 1.68·21-s − 3.77·22-s − 6.96·23-s + 24-s − 4.85·25-s − 5.19·26-s + 27-s + 1.68·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.170·5-s + 0.408·6-s + 0.638·7-s + 0.353·8-s + 0.333·9-s + 0.120·10-s − 1.13·11-s + 0.288·12-s − 1.43·13-s + 0.451·14-s + 0.0983·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.311·19-s + 0.0851·20-s + 0.368·21-s − 0.805·22-s − 1.45·23-s + 0.204·24-s − 0.971·25-s − 1.01·26-s + 0.192·27-s + 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 0.380T + 5T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 + 6.96T + 23T^{2} \) |
| 29 | \( 1 + 8.49T + 29T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 - 0.613T + 37T^{2} \) |
| 41 | \( 1 + 6.23T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 - 6.02T + 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 1.50T + 79T^{2} \) |
| 83 | \( 1 + 6.13T + 83T^{2} \) |
| 89 | \( 1 - 0.505T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.61622329753582728748726080644, −7.27025748174133492803088904658, −6.16169837584311852770101435620, −5.34752543822995999404865448887, −4.94020466304484879586374841832, −4.02521335402705904517356592335, −3.28425707247060373880870941913, −2.13512025491712521838839710011, −2.00215973052101583884117576647, 0,
2.00215973052101583884117576647, 2.13512025491712521838839710011, 3.28425707247060373880870941913, 4.02521335402705904517356592335, 4.94020466304484879586374841832, 5.34752543822995999404865448887, 6.16169837584311852770101435620, 7.27025748174133492803088904658, 7.61622329753582728748726080644
