| L(s) = 1 | − 2.07·2-s + 3-s + 2.31·4-s − 2.07·6-s − 0.151·7-s − 0.662·8-s + 9-s − 4.71·11-s + 2.31·12-s + 0.171·13-s + 0.314·14-s − 3.26·16-s + 2.52·17-s − 2.07·18-s + 7.60·19-s − 0.151·21-s + 9.79·22-s + 6.83·23-s − 0.662·24-s − 0.356·26-s + 27-s − 0.351·28-s − 9.08·29-s − 3.81·31-s + 8.10·32-s − 4.71·33-s − 5.24·34-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 0.577·3-s + 1.15·4-s − 0.848·6-s − 0.0572·7-s − 0.234·8-s + 0.333·9-s − 1.42·11-s + 0.669·12-s + 0.0475·13-s + 0.0840·14-s − 0.815·16-s + 0.611·17-s − 0.489·18-s + 1.74·19-s − 0.0330·21-s + 2.08·22-s + 1.42·23-s − 0.135·24-s − 0.0699·26-s + 0.192·27-s − 0.0663·28-s − 1.68·29-s − 0.685·31-s + 1.43·32-s − 0.820·33-s − 0.899·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 7 | \( 1 + 0.151T + 7T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 13 | \( 1 - 0.171T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 7.60T + 19T^{2} \) |
| 23 | \( 1 - 6.83T + 23T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 - 0.510T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 + 4.14T + 43T^{2} \) |
| 47 | \( 1 - 0.851T + 47T^{2} \) |
| 53 | \( 1 + 9.78T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 + 1.67T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 - 3.38T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 0.333T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 6.78T + 97T^{2} \) |
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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
−7.61716312648990287718059300319, −7.36932429334882244303430834472, −6.39539456635429631773100373737, −5.27828265834264585151810310919, −4.91748795227641271349911610051, −3.48387606402173707355990388393, −2.98512720171723330893146183626, −1.97532619383712654426232442711, −1.16709312355189143625411190007, 0,
1.16709312355189143625411190007, 1.97532619383712654426232442711, 2.98512720171723330893146183626, 3.48387606402173707355990388393, 4.91748795227641271349911610051, 5.27828265834264585151810310919, 6.39539456635429631773100373737, 7.36932429334882244303430834472, 7.61716312648990287718059300319
