| L(s) = 1 | + 2.35·2-s + 3-s + 3.53·4-s + 4.11·5-s + 2.35·6-s + 3.60·8-s + 9-s + 9.68·10-s + 11-s + 3.53·12-s − 5.39·13-s + 4.11·15-s + 1.41·16-s − 2.35·17-s + 2.35·18-s − 5.74·19-s + 14.5·20-s + 2.35·22-s + 1.41·23-s + 3.60·24-s + 11.9·25-s − 12.6·26-s + 27-s − 5.13·29-s + 9.68·30-s + 0.533·31-s − 3.87·32-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.76·4-s + 1.84·5-s + 0.960·6-s + 1.27·8-s + 0.333·9-s + 3.06·10-s + 0.301·11-s + 1.02·12-s − 1.49·13-s + 1.06·15-s + 0.354·16-s − 0.570·17-s + 0.554·18-s − 1.31·19-s + 3.25·20-s + 0.501·22-s + 0.295·23-s + 0.736·24-s + 2.38·25-s − 2.48·26-s + 0.192·27-s − 0.953·29-s + 1.76·30-s + 0.0958·31-s − 0.685·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.704721257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.704721257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 + 5.74T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 - 0.533T + 31T^{2} \) |
| 37 | \( 1 - 2.41T + 37T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 3.76T + 61T^{2} \) |
| 67 | \( 1 - 2.02T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 + 6.47T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 8.73T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
| 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
−9.443023831926039279348266759815, −8.804409999537167784870094088638, −7.42557444465504994755056897417, −6.57096047265684613971832870563, −6.10325198925872940182787732544, −5.08149367032003454223311703485, −4.62164016014607923072023493414, −3.39457802781172752804866617598, −2.31610019163407659275736506314, −2.01127534150320006847612400400,
2.01127534150320006847612400400, 2.31610019163407659275736506314, 3.39457802781172752804866617598, 4.62164016014607923072023493414, 5.08149367032003454223311703485, 6.10325198925872940182787732544, 6.57096047265684613971832870563, 7.42557444465504994755056897417, 8.804409999537167784870094088638, 9.443023831926039279348266759815
