| L(s) = 1 | − 2.64·2-s + 2.16·3-s + 5.00·4-s + 5-s − 5.73·6-s + 7-s − 7.95·8-s + 1.69·9-s − 2.64·10-s + 10.8·12-s − 3.12·13-s − 2.64·14-s + 2.16·15-s + 11.0·16-s − 5.62·17-s − 4.48·18-s + 6.54·19-s + 5.00·20-s + 2.16·21-s − 5.20·23-s − 17.2·24-s + 25-s + 8.27·26-s − 2.83·27-s + 5.00·28-s + 8.40·29-s − 5.73·30-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 1.25·3-s + 2.50·4-s + 0.447·5-s − 2.34·6-s + 0.377·7-s − 2.81·8-s + 0.564·9-s − 0.836·10-s + 3.12·12-s − 0.866·13-s − 0.707·14-s + 0.559·15-s + 2.75·16-s − 1.36·17-s − 1.05·18-s + 1.50·19-s + 1.11·20-s + 0.472·21-s − 1.08·23-s − 3.51·24-s + 0.200·25-s + 1.62·26-s − 0.544·27-s + 0.945·28-s + 1.56·29-s − 1.04·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.342201433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.342201433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 - 2.16T + 3T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 - 8.40T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 8.89T + 73T^{2} \) |
| 79 | \( 1 + 3.27T + 79T^{2} \) |
| 83 | \( 1 - 0.721T + 83T^{2} \) |
| 89 | \( 1 + 0.247T + 89T^{2} \) |
| 97 | \( 1 - 7.36T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.448947922936310818056307650102, −7.952047675114862239008258489124, −7.32813186894992207168019898928, −6.66154845132919606519124439508, −5.75101293706086705474738035727, −4.56944184945463156326914893759, −3.27256230351878579695007124987, −2.39925310158344064537776612004, −2.05166747776325286717700464167, −0.802497011457203458809216403516,
0.802497011457203458809216403516, 2.05166747776325286717700464167, 2.39925310158344064537776612004, 3.27256230351878579695007124987, 4.56944184945463156326914893759, 5.75101293706086705474738035727, 6.66154845132919606519124439508, 7.32813186894992207168019898928, 7.952047675114862239008258489124, 8.448947922936310818056307650102
