| L(s) = 1 | − 2.61·2-s + 0.399·3-s + 4.84·4-s − 5-s − 1.04·6-s − 7-s − 7.45·8-s − 2.84·9-s + 2.61·10-s + 1.74·11-s + 1.93·12-s + 4.62·13-s + 2.61·14-s − 0.399·15-s + 9.81·16-s + 5.87·17-s + 7.43·18-s + 7.96·19-s − 4.84·20-s − 0.399·21-s − 4.56·22-s − 1.11·23-s − 2.98·24-s + 25-s − 12.1·26-s − 2.33·27-s − 4.84·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.230·3-s + 2.42·4-s − 0.447·5-s − 0.427·6-s − 0.377·7-s − 2.63·8-s − 0.946·9-s + 0.827·10-s + 0.526·11-s + 0.559·12-s + 1.28·13-s + 0.699·14-s − 0.103·15-s + 2.45·16-s + 1.42·17-s + 1.75·18-s + 1.82·19-s − 1.08·20-s − 0.0872·21-s − 0.973·22-s − 0.233·23-s − 0.608·24-s + 0.200·25-s − 2.37·26-s − 0.449·27-s − 0.916·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9448832563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9448832563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.399T + 3T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 + 5.02T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 3.11T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 3.77T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 + 0.980T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 3.27T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.941498048290799842043984713300, −7.50753189307044917916872167838, −6.74626156602419467979643177477, −5.91477925767640510721903061493, −5.51228902522193476555982621732, −3.84088274714834448597884992475, −3.26600101430139874873666521296, −2.53631700269727546285715341228, −1.28341691580384698020701666927, −0.72713223109038958405394373507,
0.72713223109038958405394373507, 1.28341691580384698020701666927, 2.53631700269727546285715341228, 3.26600101430139874873666521296, 3.84088274714834448597884992475, 5.51228902522193476555982621732, 5.91477925767640510721903061493, 6.74626156602419467979643177477, 7.50753189307044917916872167838, 7.941498048290799842043984713300
