| L(s) = 1 | − 1.90·2-s + 0.192·3-s + 1.63·4-s + 0.483·5-s − 0.366·6-s + 3.15·7-s + 0.704·8-s − 2.96·9-s − 0.921·10-s + 0.275·11-s + 0.313·12-s − 1.15·13-s − 6.01·14-s + 0.0930·15-s − 4.60·16-s − 4.88·17-s + 5.64·18-s − 19-s + 0.788·20-s + 0.607·21-s − 0.525·22-s + 7.55·23-s + 0.135·24-s − 4.76·25-s + 2.20·26-s − 1.14·27-s + 5.14·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.111·3-s + 0.815·4-s + 0.216·5-s − 0.149·6-s + 1.19·7-s + 0.249·8-s − 0.987·9-s − 0.291·10-s + 0.0832·11-s + 0.0905·12-s − 0.321·13-s − 1.60·14-s + 0.0240·15-s − 1.15·16-s − 1.18·17-s + 1.33·18-s − 0.229·19-s + 0.176·20-s + 0.132·21-s − 0.112·22-s + 1.57·23-s + 0.0276·24-s − 0.953·25-s + 0.433·26-s − 0.220·27-s + 0.971·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 3 | \( 1 - 0.192T + 3T^{2} \) |
| 5 | \( 1 - 0.483T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 0.275T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 - 8.37T + 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 + 4.39T + 53T^{2} \) |
| 59 | \( 1 - 2.49T + 59T^{2} \) |
| 61 | \( 1 - 8.52T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 - 8.31T + 73T^{2} \) |
| 79 | \( 1 + 1.19T + 79T^{2} \) |
| 83 | \( 1 + 4.01T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 3.23T + 97T^{2} \) |
| 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
−8.101276313480150221330288150412, −7.24547697866697237572654197439, −6.58395584391994109523580461916, −5.61543306920947170934614706765, −4.84007268043344802582511495706, −4.18326249147111962540206587396, −2.78649849728044321013664993230, −2.09698110094489776488385919025, −1.20282520903872912930234141402, 0,
1.20282520903872912930234141402, 2.09698110094489776488385919025, 2.78649849728044321013664993230, 4.18326249147111962540206587396, 4.84007268043344802582511495706, 5.61543306920947170934614706765, 6.58395584391994109523580461916, 7.24547697866697237572654197439, 8.101276313480150221330288150412
