| L(s) = 1 | + 2.04·3-s + 3.17·5-s + 1.16·9-s + 2.64·11-s + 5.60·13-s + 6.48·15-s + 7.48·17-s − 19-s + 4.87·23-s + 5.09·25-s − 3.74·27-s + 2.60·29-s − 3.41·31-s + 5.39·33-s − 9.05·37-s + 11.4·39-s − 10.4·41-s + 3.77·43-s + 3.70·45-s + 7.21·47-s + 15.2·51-s − 9.74·53-s + 8.39·55-s − 2.04·57-s + 3.30·59-s − 2.35·61-s + 17.7·65-s + ⋯ |
| L(s) = 1 | + 1.17·3-s + 1.42·5-s + 0.389·9-s + 0.796·11-s + 1.55·13-s + 1.67·15-s + 1.81·17-s − 0.229·19-s + 1.01·23-s + 1.01·25-s − 0.719·27-s + 0.484·29-s − 0.612·31-s + 0.938·33-s − 1.48·37-s + 1.83·39-s − 1.62·41-s + 0.575·43-s + 0.552·45-s + 1.05·47-s + 2.13·51-s − 1.33·53-s + 1.13·55-s − 0.270·57-s + 0.430·59-s − 0.301·61-s + 2.20·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.195184070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.195184070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 - 3.30T + 59T^{2} \) |
| 61 | \( 1 + 2.35T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 8.57T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 + 6.51T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 7.06T + 97T^{2} \) |
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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
−8.184123498515118066912100537527, −7.14478996066963123021458571394, −6.55449264172348129969144627031, −5.68758148742881210304473925446, −5.37131210403819024240035216620, −4.04353688401891213443012804667, −3.36999502093772818612884992383, −2.81327218714800778804683842661, −1.66597776783543413463773721574, −1.29523553213946620977412569662,
1.29523553213946620977412569662, 1.66597776783543413463773721574, 2.81327218714800778804683842661, 3.36999502093772818612884992383, 4.04353688401891213443012804667, 5.37131210403819024240035216620, 5.68758148742881210304473925446, 6.55449264172348129969144627031, 7.14478996066963123021458571394, 8.184123498515118066912100537527
