| L(s) = 1 | + 3-s − 4.20·5-s + 3.93·7-s + 9-s − 2.04·11-s + 2.48·13-s − 4.20·15-s − 7.89·17-s − 4.59·19-s + 3.93·21-s + 3.05·23-s + 12.6·25-s + 27-s + 10.2·29-s − 4.68·31-s − 2.04·33-s − 16.5·35-s + 9.76·37-s + 2.48·39-s − 7.20·41-s + 4.73·43-s − 4.20·45-s − 8.18·47-s + 8.47·49-s − 7.89·51-s − 6.41·53-s + 8.57·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.87·5-s + 1.48·7-s + 0.333·9-s − 0.615·11-s + 0.689·13-s − 1.08·15-s − 1.91·17-s − 1.05·19-s + 0.858·21-s + 0.636·23-s + 2.53·25-s + 0.192·27-s + 1.90·29-s − 0.842·31-s − 0.355·33-s − 2.79·35-s + 1.60·37-s + 0.398·39-s − 1.12·41-s + 0.721·43-s − 0.626·45-s − 1.19·47-s + 1.21·49-s − 1.10·51-s − 0.881·53-s + 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 19 | \( 1 + 4.59T + 19T^{2} \) |
| 23 | \( 1 - 3.05T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 - 9.76T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 + 8.18T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 - 9.60T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 - 0.838T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 8.17T + 83T^{2} \) |
| 89 | \( 1 - 9.03T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
| 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
−7.71578132357623039007841190215, −6.99566562515375880324165796347, −6.33708293898263240389329917559, −4.95053945056398758950657224298, −4.50805824998019564229262272005, −4.13055215536065902543450421362, −3.12303997740158630840899443709, −2.33578427482627606553716208129, −1.24416718341601340459059127255, 0,
1.24416718341601340459059127255, 2.33578427482627606553716208129, 3.12303997740158630840899443709, 4.13055215536065902543450421362, 4.50805824998019564229262272005, 4.95053945056398758950657224298, 6.33708293898263240389329917559, 6.99566562515375880324165796347, 7.71578132357623039007841190215
