| L(s) = 1 | − 0.571·2-s − 1.67·4-s − 1.42·7-s + 2.10·8-s + 2.67·11-s − 4.67·13-s + 0.816·14-s + 2.14·16-s − 2.67·17-s + 4.67·19-s − 1.52·22-s + 5.91·23-s + 2.67·26-s + 2.38·28-s − 9.48·29-s + 6.96·31-s − 5.42·32-s + 1.52·34-s + 1.81·37-s − 2.67·38-s − 1.47·41-s − 0.471·43-s − 4.47·44-s − 3.38·46-s − 6.95·47-s − 4.96·49-s + 7.81·52-s + ⋯ |
| L(s) = 1 | − 0.404·2-s − 0.836·4-s − 0.539·7-s + 0.742·8-s + 0.805·11-s − 1.29·13-s + 0.218·14-s + 0.535·16-s − 0.648·17-s + 1.07·19-s − 0.325·22-s + 1.23·23-s + 0.524·26-s + 0.451·28-s − 1.76·29-s + 1.25·31-s − 0.959·32-s + 0.262·34-s + 0.298·37-s − 0.433·38-s − 0.229·41-s − 0.0718·43-s − 0.674·44-s − 0.499·46-s − 1.01·47-s − 0.708·49-s + 1.08·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.571T + 2T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 - 5.91T + 23T^{2} \) |
| 29 | \( 1 + 9.48T + 29T^{2} \) |
| 31 | \( 1 - 6.96T + 31T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 + 0.471T + 43T^{2} \) |
| 47 | \( 1 + 6.95T + 47T^{2} \) |
| 53 | \( 1 - 1.14T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 - 6.59T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.71T + 73T^{2} \) |
| 79 | \( 1 + 0.287T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 7.83T + 97T^{2} \) |
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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
−9.000764897164661321223693345571, −8.022004276199081370483022612934, −7.26999283100990533140522291595, −6.56253615610313904704712307875, −5.38858620420559787883300773927, −4.73197584440309450651301583949, −3.80781356672708679629450073798, −2.81516137790516474252937843148, −1.35914631696800017849992079384, 0,
1.35914631696800017849992079384, 2.81516137790516474252937843148, 3.80781356672708679629450073798, 4.73197584440309450651301583949, 5.38858620420559787883300773927, 6.56253615610313904704712307875, 7.26999283100990533140522291595, 8.022004276199081370483022612934, 9.000764897164661321223693345571
