L(s) = 1 | + 2-s + 2.82·3-s + 4-s + 2.82·6-s + 8-s + 5.00·9-s − 11-s + 2.82·12-s + 4.24·13-s + 16-s − 2.82·17-s + 5.00·18-s − 4.24·19-s − 22-s + 6·23-s + 2.82·24-s − 5·25-s + 4.24·26-s + 5.65·27-s − 4·29-s − 7.07·31-s + 32-s − 2.82·33-s − 2.82·34-s + 5.00·36-s + 2·37-s − 4.24·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.63·3-s + 0.5·4-s + 1.15·6-s + 0.353·8-s + 1.66·9-s − 0.301·11-s + 0.816·12-s + 1.17·13-s + 0.250·16-s − 0.685·17-s + 1.17·18-s − 0.973·19-s − 0.213·22-s + 1.25·23-s + 0.577·24-s − 25-s + 0.832·26-s + 1.08·27-s − 0.742·29-s − 1.27·31-s + 0.176·32-s − 0.492·33-s − 0.485·34-s + 0.833·36-s + 0.328·37-s − 0.688·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.265042220\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.265042220\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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
−9.664468343642505342977517911174, −8.942429623432213009222951527686, −8.274134519278857735927129108357, −7.47378900406172545797151789958, −6.60264138382635230738572977583, −5.54323956136095152735740092847, −4.24751455046883416199796491108, −3.65613553033659939715923863264, −2.66257201384197164240263590851, −1.74212596595951751211122154111,
1.74212596595951751211122154111, 2.66257201384197164240263590851, 3.65613553033659939715923863264, 4.24751455046883416199796491108, 5.54323956136095152735740092847, 6.60264138382635230738572977583, 7.47378900406172545797151789958, 8.274134519278857735927129108357, 8.942429623432213009222951527686, 9.664468343642505342977517911174