| L(s) = 1 | + 2-s + 0.232·3-s + 4-s + 0.663·5-s + 0.232·6-s + 8-s − 2.94·9-s + 0.663·10-s + 0.431·11-s + 0.232·12-s + 1.71·13-s + 0.154·15-s + 16-s − 3.55·17-s − 2.94·18-s − 7.71·19-s + 0.663·20-s + 0.431·22-s + 0.386·23-s + 0.232·24-s − 4.55·25-s + 1.71·26-s − 1.38·27-s − 6.37·29-s + 0.154·30-s − 5.94·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.134·3-s + 0.5·4-s + 0.296·5-s + 0.0948·6-s + 0.353·8-s − 0.982·9-s + 0.209·10-s + 0.130·11-s + 0.0670·12-s + 0.475·13-s + 0.0398·15-s + 0.250·16-s − 0.863·17-s − 0.694·18-s − 1.76·19-s + 0.148·20-s + 0.0919·22-s + 0.0805·23-s + 0.0474·24-s − 0.911·25-s + 0.336·26-s − 0.265·27-s − 1.18·29-s + 0.0281·30-s − 1.06·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.232T + 3T^{2} \) |
| 5 | \( 1 - 0.663T + 5T^{2} \) |
| 11 | \( 1 - 0.431T + 11T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 23 | \( 1 - 0.386T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 + 5.94T + 31T^{2} \) |
| 37 | \( 1 + 1.53T + 37T^{2} \) |
| 43 | \( 1 - 2.61T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + 0.509T + 59T^{2} \) |
| 61 | \( 1 + 7.99T + 61T^{2} \) |
| 67 | \( 1 + 3.36T + 67T^{2} \) |
| 71 | \( 1 - 0.518T + 71T^{2} \) |
| 73 | \( 1 - 7.42T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 3.95T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
| 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
−8.111643457273058507480120367354, −7.22477041422119635944426828877, −6.35664994178247612274208320519, −5.90923400669251891520754163609, −5.14118583812174923711477429135, −4.14440211500312221512282078661, −3.57131987621155125751375573583, −2.43962137553585885858118387810, −1.83832732495287979125839646767, 0,
1.83832732495287979125839646767, 2.43962137553585885858118387810, 3.57131987621155125751375573583, 4.14440211500312221512282078661, 5.14118583812174923711477429135, 5.90923400669251891520754163609, 6.35664994178247612274208320519, 7.22477041422119635944426828877, 8.111643457273058507480120367354
