| L(s) = 1 | + 2-s − 1.17·3-s + 4-s − 1.17·6-s − 2.08·7-s + 8-s − 1.61·9-s − 2.99·11-s − 1.17·12-s − 2.08·14-s + 16-s − 1.26·17-s − 1.61·18-s − 3.89·19-s + 2.45·21-s − 2.99·22-s − 0.899·23-s − 1.17·24-s + 5.43·27-s − 2.08·28-s − 0.470·29-s − 0.277·31-s + 32-s + 3.52·33-s − 1.26·34-s − 1.61·36-s + 7.03·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.679·3-s + 0.5·4-s − 0.480·6-s − 0.788·7-s + 0.353·8-s − 0.538·9-s − 0.902·11-s − 0.339·12-s − 0.557·14-s + 0.250·16-s − 0.305·17-s − 0.380·18-s − 0.892·19-s + 0.535·21-s − 0.637·22-s − 0.187·23-s − 0.240·24-s + 1.04·27-s − 0.394·28-s − 0.0874·29-s − 0.0497·31-s + 0.176·32-s + 0.612·33-s − 0.216·34-s − 0.269·36-s + 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188279835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188279835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.17T + 3T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 + 0.899T + 23T^{2} \) |
| 29 | \( 1 + 0.470T + 29T^{2} \) |
| 31 | \( 1 + 0.277T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 7.78T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 7.64T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 6.91T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
| 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
−7.80379728418566593603615221013, −6.71564355322836678518006055948, −6.32151699368646826917782173850, −5.79469710439781645878344895288, −4.97834306598937519532772901532, −4.47744429518278229698245648675, −3.42135035644133031985681773251, −2.82505257962639301020585379677, −1.96737691703227052298493723139, −0.46532637909065311232229640695,
0.46532637909065311232229640695, 1.96737691703227052298493723139, 2.82505257962639301020585379677, 3.42135035644133031985681773251, 4.47744429518278229698245648675, 4.97834306598937519532772901532, 5.79469710439781645878344895288, 6.32151699368646826917782173850, 6.71564355322836678518006055948, 7.80379728418566593603615221013
