| L(s) = 1 | − 1.86·2-s − 2.63·3-s + 1.49·4-s + 5-s + 4.92·6-s − 2.31·7-s + 0.944·8-s + 3.92·9-s − 1.86·10-s + 11-s − 3.93·12-s − 2.05·13-s + 4.33·14-s − 2.63·15-s − 4.75·16-s − 5.64·17-s − 7.34·18-s − 4.69·19-s + 1.49·20-s + 6.10·21-s − 1.86·22-s − 7.33·23-s − 2.48·24-s + 25-s + 3.84·26-s − 2.44·27-s − 3.46·28-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 1.51·3-s + 0.747·4-s + 0.447·5-s + 2.00·6-s − 0.876·7-s + 0.333·8-s + 1.30·9-s − 0.591·10-s + 0.301·11-s − 1.13·12-s − 0.569·13-s + 1.15·14-s − 0.679·15-s − 1.18·16-s − 1.36·17-s − 1.73·18-s − 1.07·19-s + 0.334·20-s + 1.33·21-s − 0.398·22-s − 1.52·23-s − 0.507·24-s + 0.200·25-s + 0.753·26-s − 0.470·27-s − 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08029677088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08029677088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 3 | \( 1 + 2.63T + 3T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 + 5.64T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.817T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 + 8.22T + 53T^{2} \) |
| 59 | \( 1 + 0.531T + 59T^{2} \) |
| 61 | \( 1 + 6.90T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 8.72T + 71T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 7.28T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.596678570944753738088436015109, −7.68462239038853836448382355888, −6.89608622508331053226142709909, −6.29022851728662646478578290515, −5.89045799476414159724235808270, −4.67855218643994933376482341875, −4.15970695568918112618680780974, −2.49440625847153873078316536192, −1.57484530443606877549697352810, −0.21435295700147114082587433837,
0.21435295700147114082587433837, 1.57484530443606877549697352810, 2.49440625847153873078316536192, 4.15970695568918112618680780974, 4.67855218643994933376482341875, 5.89045799476414159724235808270, 6.29022851728662646478578290515, 6.89608622508331053226142709909, 7.68462239038853836448382355888, 8.596678570944753738088436015109
