| L(s) = 1 | + 2-s − 1.41·3-s + 4-s − 1.41·5-s − 1.41·6-s + 8-s − 0.999·9-s − 1.41·10-s − 2·11-s − 1.41·12-s + 4.24·13-s + 2.00·15-s + 16-s + 2.82·17-s − 0.999·18-s − 3.74·19-s − 1.41·20-s − 2·22-s + 3.29·23-s − 1.41·24-s − 2.99·25-s + 4.24·26-s + 5.65·27-s − 8.93·29-s + 2.00·30-s + 6.06·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.816·3-s + 0.5·4-s − 0.632·5-s − 0.577·6-s + 0.353·8-s − 0.333·9-s − 0.447·10-s − 0.603·11-s − 0.408·12-s + 1.17·13-s + 0.516·15-s + 0.250·16-s + 0.685·17-s − 0.235·18-s − 0.858·19-s − 0.316·20-s − 0.426·22-s + 0.686·23-s − 0.288·24-s − 0.599·25-s + 0.832·26-s + 1.08·27-s − 1.65·29-s + 0.365·30-s + 1.09·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 - 3.29T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 + 5.64T + 37T^{2} \) |
| 41 | \( 1 - 7.98T + 41T^{2} \) |
| 43 | \( 1 + 0.354T + 43T^{2} \) |
| 47 | \( 1 - 9.81T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 8.39T + 59T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 6.57T + 73T^{2} \) |
| 83 | \( 1 + 3.74T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
| 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
−7.47334142404718712357748113023, −6.58493682620652699288123995887, −5.97895470493932456177916031363, −5.50037815288032664175411664762, −4.75158115413131743754305433092, −3.94852760250242601714691283645, −3.33275634426814096249125445733, −2.40301732934167237663069288081, −1.18638467110435038784685531017, 0,
1.18638467110435038784685531017, 2.40301732934167237663069288081, 3.33275634426814096249125445733, 3.94852760250242601714691283645, 4.75158115413131743754305433092, 5.50037815288032664175411664762, 5.97895470493932456177916031363, 6.58493682620652699288123995887, 7.47334142404718712357748113023
