| L(s) = 1 | − 1.22·2-s − 3-s − 0.490·4-s + 1.22·6-s + 3.05·8-s + 9-s − 4.74·11-s + 0.490·12-s − 6.25·13-s − 2.77·16-s − 6.28·17-s − 1.22·18-s − 4.94·19-s + 5.83·22-s − 4.98·23-s − 3.05·24-s + 7.68·26-s − 27-s + 4·29-s − 1.65·31-s − 2.70·32-s + 4.74·33-s + 7.72·34-s − 0.490·36-s + 4.43·37-s + 6.07·38-s + 6.25·39-s + ⋯ |
| L(s) = 1 | − 0.868·2-s − 0.577·3-s − 0.245·4-s + 0.501·6-s + 1.08·8-s + 0.333·9-s − 1.43·11-s + 0.141·12-s − 1.73·13-s − 0.694·16-s − 1.52·17-s − 0.289·18-s − 1.13·19-s + 1.24·22-s − 1.03·23-s − 0.624·24-s + 1.50·26-s − 0.192·27-s + 0.742·29-s − 0.297·31-s − 0.478·32-s + 0.826·33-s + 1.32·34-s − 0.0817·36-s + 0.729·37-s + 0.986·38-s + 1.00·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07693332238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07693332238\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 + 6.28T + 17T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 3.76T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 59 | \( 1 + 0.234T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 2.49T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 0.831T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 + 2.28T + 89T^{2} \) |
| 97 | \( 1 + 0.476T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.424730248790285693825941676802, −7.958566866800647626179228278392, −7.17947873567928603770051812932, −6.50469235866939394681008665717, −5.40988605961734331447787056191, −4.72517333060376803476830940823, −4.24727286286229418491598984029, −2.62436325602347059777791962325, −1.90976916441416359320012703533, −0.17966654048171228535908203735,
0.17966654048171228535908203735, 1.90976916441416359320012703533, 2.62436325602347059777791962325, 4.24727286286229418491598984029, 4.72517333060376803476830940823, 5.40988605961734331447787056191, 6.50469235866939394681008665717, 7.17947873567928603770051812932, 7.958566866800647626179228278392, 8.424730248790285693825941676802
