| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s + 5·5-s − 8·6-s − 20·7-s + 8·8-s − 11·9-s + 10·10-s − 44·11-s − 16·12-s + 42·13-s − 40·14-s − 20·15-s + 16·16-s − 86·17-s − 22·18-s + 19·19-s + 20·20-s + 80·21-s − 88·22-s − 164·23-s − 32·24-s + 25·25-s + 84·26-s + 152·27-s − 80·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s + 0.447·5-s − 0.544·6-s − 1.07·7-s + 0.353·8-s − 0.407·9-s + 0.316·10-s − 1.20·11-s − 0.384·12-s + 0.896·13-s − 0.763·14-s − 0.344·15-s + 1/4·16-s − 1.22·17-s − 0.288·18-s + 0.229·19-s + 0.223·20-s + 0.831·21-s − 0.852·22-s − 1.48·23-s − 0.272·24-s + 1/5·25-s + 0.633·26-s + 1.08·27-s − 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 86 T + p^{3} T^{2} \) |
| 23 | \( 1 + 164 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 312 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 34 T + p^{3} T^{2} \) |
| 43 | \( 1 + 432 T + p^{3} T^{2} \) |
| 47 | \( 1 - 580 T + p^{3} T^{2} \) |
| 53 | \( 1 - 506 T + p^{3} T^{2} \) |
| 59 | \( 1 - 364 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 924 T + p^{3} T^{2} \) |
| 71 | \( 1 - 320 T + p^{3} T^{2} \) |
| 73 | \( 1 + 542 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1208 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1120 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1022 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1166 T + p^{3} T^{2} \) |
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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
−11.58612399691585644828569213200, −10.84095871809509681032446392806, −9.899793101950890731014351743848, −8.578136134646510593071588527285, −7.05777051964957390216058301922, −5.98878204292262418958489604657, −5.45977410423522683416261940185, −3.84282973112482394919715116660, −2.40294405515677059811414395528, 0,
2.40294405515677059811414395528, 3.84282973112482394919715116660, 5.45977410423522683416261940185, 5.98878204292262418958489604657, 7.05777051964957390216058301922, 8.578136134646510593071588527285, 9.899793101950890731014351743848, 10.84095871809509681032446392806, 11.58612399691585644828569213200
