| L(s) = 1 | − 0.766·2-s + 3-s − 1.41·4-s + 0.561·5-s − 0.766·6-s + 2.61·8-s + 9-s − 0.430·10-s − 3.11·11-s − 1.41·12-s + 5.92·13-s + 0.561·15-s + 0.823·16-s + 3.15·17-s − 0.766·18-s − 0.142·19-s − 0.793·20-s + 2.38·22-s − 4.62·23-s + 2.61·24-s − 4.68·25-s − 4.53·26-s + 27-s − 8.30·29-s − 0.430·30-s − 0.336·31-s − 5.86·32-s + ⋯ |
| L(s) = 1 | − 0.541·2-s + 0.577·3-s − 0.706·4-s + 0.251·5-s − 0.312·6-s + 0.924·8-s + 0.333·9-s − 0.136·10-s − 0.938·11-s − 0.407·12-s + 1.64·13-s + 0.145·15-s + 0.205·16-s + 0.764·17-s − 0.180·18-s − 0.0326·19-s − 0.177·20-s + 0.508·22-s − 0.965·23-s + 0.533·24-s − 0.936·25-s − 0.890·26-s + 0.192·27-s − 1.54·29-s − 0.0785·30-s − 0.0604·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.766T + 2T^{2} \) |
| 5 | \( 1 - 0.561T + 5T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 5.92T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 0.142T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 + 0.336T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 43 | \( 1 - 9.28T + 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 - 0.911T + 59T^{2} \) |
| 61 | \( 1 + 3.48T + 61T^{2} \) |
| 67 | \( 1 + 0.537T + 67T^{2} \) |
| 71 | \( 1 + 0.835T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 5.99T + 83T^{2} \) |
| 89 | \( 1 - 18.7T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
| 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
−7.81913703836580628407827056002, −7.46235936974584055248380322451, −6.19740097369008641829403652728, −5.63106457784084258262873770745, −4.83005896415358642133314206341, −3.77961256319912985355771299643, −3.46257277494936833167249644042, −2.09032167829964142852757543338, −1.34859254131930283217143271422, 0,
1.34859254131930283217143271422, 2.09032167829964142852757543338, 3.46257277494936833167249644042, 3.77961256319912985355771299643, 4.83005896415358642133314206341, 5.63106457784084258262873770745, 6.19740097369008641829403652728, 7.46235936974584055248380322451, 7.81913703836580628407827056002
