| L(s) = 1 | − 1.26·2-s − 0.395·4-s − 4.47·5-s + 4.92·7-s + 3.03·8-s + 5.66·10-s − 6.23·14-s − 3.05·16-s + 5.19·19-s + 1.76·20-s + 14.9·25-s − 1.94·28-s − 2.20·32-s − 22.0·35-s − 6.57·38-s − 13.5·40-s + 8.34·41-s + 11.1·47-s + 17.2·49-s − 18.9·50-s + 14.9·56-s − 3.13·59-s + 8.89·64-s + 12·67-s + 27.8·70-s + 15.9·71-s − 2.05·76-s + ⋯ |
| L(s) = 1 | − 0.895·2-s − 0.197·4-s − 1.99·5-s + 1.86·7-s + 1.07·8-s + 1.79·10-s − 1.66·14-s − 0.763·16-s + 1.19·19-s + 0.395·20-s + 2.99·25-s − 0.367·28-s − 0.389·32-s − 3.72·35-s − 1.06·38-s − 2.14·40-s + 1.30·41-s + 1.62·47-s + 2.46·49-s − 2.68·50-s + 1.99·56-s − 0.407·59-s + 1.11·64-s + 1.46·67-s + 3.33·70-s + 1.89·71-s − 0.235·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.018306906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018306906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 5 | \( 1 + 4.47T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 3.13T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 9.44T + 97T^{2} \) |
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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
−7.88234463471487353022600583246, −7.51496153052595719276021501822, −6.90363842008315633811595243516, −5.38454218062288828623016667985, −4.90418167637848647755588968531, −4.19301305955767429058401770261, −3.76720803246034656145523526416, −2.51948350457426101489188997167, −1.26222541268488473533455372148, −0.67861781184015970409073640383,
0.67861781184015970409073640383, 1.26222541268488473533455372148, 2.51948350457426101489188997167, 3.76720803246034656145523526416, 4.19301305955767429058401770261, 4.90418167637848647755588968531, 5.38454218062288828623016667985, 6.90363842008315633811595243516, 7.51496153052595719276021501822, 7.88234463471487353022600583246
