| L(s) = 1 | − 2.30·2-s + 3-s + 3.30·4-s + 5-s − 2.30·6-s − 4.60·7-s − 3.00·8-s − 2·9-s − 2.30·10-s + 4.60·11-s + 3.30·12-s + 5.60·13-s + 10.6·14-s + 15-s + 0.302·16-s − 6·17-s + 4.60·18-s − 1.39·19-s + 3.30·20-s − 4.60·21-s − 10.6·22-s − 3.00·24-s + 25-s − 12.9·26-s − 5·27-s − 15.2·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.577·3-s + 1.65·4-s + 0.447·5-s − 0.940·6-s − 1.74·7-s − 1.06·8-s − 0.666·9-s − 0.728·10-s + 1.38·11-s + 0.953·12-s + 1.55·13-s + 2.83·14-s + 0.258·15-s + 0.0756·16-s − 1.45·17-s + 1.08·18-s − 0.319·19-s + 0.738·20-s − 1.00·21-s − 2.26·22-s − 0.612·24-s + 0.200·25-s − 2.53·26-s − 0.962·27-s − 2.87·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 1.39T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 1.39T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 7.60T + 71T^{2} \) |
| 73 | \( 1 + 6.81T + 73T^{2} \) |
| 79 | \( 1 - 1.39T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 - 4.60T + 97T^{2} \) |
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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
−8.824041525645814295214264365726, −8.073866553469656319049343189477, −6.95767373754170976804440551036, −6.30616647112097758804584349587, −6.10626323762194773191139606506, −4.16719433252599394712842281587, −3.30669147853294286933562154198, −2.41790438710254186587398873050, −1.33546866514215953554773787427, 0,
1.33546866514215953554773787427, 2.41790438710254186587398873050, 3.30669147853294286933562154198, 4.16719433252599394712842281587, 6.10626323762194773191139606506, 6.30616647112097758804584349587, 6.95767373754170976804440551036, 8.073866553469656319049343189477, 8.824041525645814295214264365726
