| L(s) = 1 | + 2-s − 2.55·3-s + 4-s − 2.55·6-s + 1.86·7-s + 8-s + 3.51·9-s + 4.42·11-s − 2.55·12-s + 1.86·14-s + 16-s − 4.16·17-s + 3.51·18-s + 4.52·19-s − 4.77·21-s + 4.42·22-s − 7.56·23-s − 2.55·24-s − 1.30·27-s + 1.86·28-s + 3.22·29-s − 8.51·31-s + 32-s − 11.3·33-s − 4.16·34-s + 3.51·36-s − 0.488·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.47·3-s + 0.5·4-s − 1.04·6-s + 0.706·7-s + 0.353·8-s + 1.17·9-s + 1.33·11-s − 0.736·12-s + 0.499·14-s + 0.250·16-s − 1.01·17-s + 0.827·18-s + 1.03·19-s − 1.04·21-s + 0.944·22-s − 1.57·23-s − 0.520·24-s − 0.251·27-s + 0.353·28-s + 0.598·29-s − 1.52·31-s + 0.176·32-s − 1.96·33-s − 0.714·34-s + 0.585·36-s − 0.0802·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.55T + 3T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 - 4.52T + 19T^{2} \) |
| 23 | \( 1 + 7.56T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 + 0.488T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 4.38T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 0.191T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 5.69T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.10T + 73T^{2} \) |
| 79 | \( 1 + 2.33T + 79T^{2} \) |
| 83 | \( 1 - 2.49T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.91T + 97T^{2} \) |
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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
−7.10455014536139249202792858025, −6.46559675863273228275647768463, −6.16379309256430685519357735659, −5.13502994554484611897563676354, −4.94717479743569527712159228204, −4.04702127307803262199979086696, −3.40848342217022323579556414639, −1.95231679079996505246680330474, −1.35878247933511454502544234410, 0,
1.35878247933511454502544234410, 1.95231679079996505246680330474, 3.40848342217022323579556414639, 4.04702127307803262199979086696, 4.94717479743569527712159228204, 5.13502994554484611897563676354, 6.16379309256430685519357735659, 6.46559675863273228275647768463, 7.10455014536139249202792858025
