| L(s) = 1 | + 2.80·2-s − 1.73·3-s + 5.84·4-s − 4.85·6-s + 7-s + 10.7·8-s − 0.000289·9-s + 1.96·11-s − 10.1·12-s + 3.06·13-s + 2.80·14-s + 18.5·16-s − 5.01·17-s − 0.000811·18-s − 0.164·19-s − 1.73·21-s + 5.51·22-s + 23-s − 18.6·24-s + 8.57·26-s + 5.19·27-s + 5.84·28-s − 1.60·29-s − 1.84·31-s + 30.3·32-s − 3.40·33-s − 14.0·34-s + ⋯ |
| L(s) = 1 | + 1.98·2-s − 0.999·3-s + 2.92·4-s − 1.98·6-s + 0.377·7-s + 3.81·8-s − 9.65e − 5·9-s + 0.593·11-s − 2.92·12-s + 0.849·13-s + 0.748·14-s + 4.62·16-s − 1.21·17-s − 0.000191·18-s − 0.0377·19-s − 0.377·21-s + 1.17·22-s + 0.208·23-s − 3.81·24-s + 1.68·26-s + 1.00·27-s + 1.10·28-s − 0.297·29-s − 0.330·31-s + 5.35·32-s − 0.593·33-s − 2.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.939348336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.939348336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 - 1.96T + 11T^{2} \) |
| 13 | \( 1 - 3.06T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 19 | \( 1 + 0.164T + 19T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 + 4.25T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 + 7.76T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 8.46T + 89T^{2} \) |
| 97 | \( 1 - 4.55T + 97T^{2} \) |
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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
−8.142925100286129885191322531469, −7.22289879373676693382231970409, −6.56133476615995286013154320223, −5.98283831656277197067867845982, −5.54861968279721757571162931030, −4.54268249463318010014394932057, −4.24834089948250348695488889918, −3.20962813498605741500163273008, −2.26039522819372306877901065916, −1.18713443169925525527359219889,
1.18713443169925525527359219889, 2.26039522819372306877901065916, 3.20962813498605741500163273008, 4.24834089948250348695488889918, 4.54268249463318010014394932057, 5.54861968279721757571162931030, 5.98283831656277197067867845982, 6.56133476615995286013154320223, 7.22289879373676693382231970409, 8.142925100286129885191322531469
