| L(s) = 1 | + 1.79·2-s − 1.59·3-s + 1.23·4-s + 5-s − 2.85·6-s − 7-s − 1.37·8-s − 0.471·9-s + 1.79·10-s − 2.88·11-s − 1.96·12-s − 4.52·13-s − 1.79·14-s − 1.59·15-s − 4.94·16-s + 6.65·17-s − 0.846·18-s − 7.03·19-s + 1.23·20-s + 1.59·21-s − 5.18·22-s + 23-s + 2.19·24-s + 25-s − 8.13·26-s + 5.51·27-s − 1.23·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 0.918·3-s + 0.616·4-s + 0.447·5-s − 1.16·6-s − 0.377·7-s − 0.487·8-s − 0.157·9-s + 0.568·10-s − 0.869·11-s − 0.565·12-s − 1.25·13-s − 0.480·14-s − 0.410·15-s − 1.23·16-s + 1.61·17-s − 0.199·18-s − 1.61·19-s + 0.275·20-s + 0.347·21-s − 1.10·22-s + 0.208·23-s + 0.447·24-s + 0.200·25-s − 1.59·26-s + 1.06·27-s − 0.232·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 3 | \( 1 + 1.59T + 3T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 + 7.03T + 19T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 2.06T + 47T^{2} \) |
| 53 | \( 1 + 6.93T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 - 3.35T + 61T^{2} \) |
| 67 | \( 1 + 9.73T + 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 8.21T + 83T^{2} \) |
| 89 | \( 1 - 1.95T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−10.09340671243310332923834835730, −9.151991987206591858483893806890, −7.924609339178187333020361641140, −6.78069623486464893408321526590, −5.97134779973014552341292888861, −5.32489887550902019330935770669, −4.72779043398416393452916477108, −3.38966648073203536545892145773, −2.38236081334714665157578085838, 0,
2.38236081334714665157578085838, 3.38966648073203536545892145773, 4.72779043398416393452916477108, 5.32489887550902019330935770669, 5.97134779973014552341292888861, 6.78069623486464893408321526590, 7.924609339178187333020361641140, 9.151991987206591858483893806890, 10.09340671243310332923834835730
