| L(s) = 1 | − 2.01·2-s + 3.02·3-s + 2.05·4-s − 6.09·6-s + 0.369·7-s − 0.110·8-s + 6.15·9-s − 1.74·11-s + 6.21·12-s + 1.11·13-s − 0.745·14-s − 3.88·16-s + 5.48·17-s − 12.3·18-s − 3.75·19-s + 1.11·21-s + 3.51·22-s + 7.24·23-s − 0.334·24-s − 2.24·26-s + 9.54·27-s + 0.760·28-s + 4.19·29-s + 0.305·31-s + 8.04·32-s − 5.28·33-s − 11.0·34-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.74·3-s + 1.02·4-s − 2.48·6-s + 0.139·7-s − 0.0390·8-s + 2.05·9-s − 0.526·11-s + 1.79·12-s + 0.309·13-s − 0.199·14-s − 0.971·16-s + 1.33·17-s − 2.92·18-s − 0.860·19-s + 0.244·21-s + 0.749·22-s + 1.51·23-s − 0.0682·24-s − 0.440·26-s + 1.83·27-s + 0.143·28-s + 0.778·29-s + 0.0549·31-s + 1.42·32-s − 0.919·33-s − 1.89·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.390494921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390494921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 2.01T + 2T^{2} \) |
| 3 | \( 1 - 3.02T + 3T^{2} \) |
| 7 | \( 1 - 0.369T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 - 5.48T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 0.305T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 - 7.17T + 43T^{2} \) |
| 47 | \( 1 - 0.810T + 47T^{2} \) |
| 53 | \( 1 + 3.91T + 53T^{2} \) |
| 59 | \( 1 + 1.85T + 59T^{2} \) |
| 61 | \( 1 - 9.68T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 - 7.13T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−10.26726801923786606480056160288, −9.498977342807506019120465663396, −8.778239798398161218146474908086, −8.219926020571777198935681583159, −7.58641610201656756020224939661, −6.73413114950597946664975841638, −4.92295768648648171557765776395, −3.55157884318988481737082954886, −2.51221127919791218134376952725, −1.33065236626948887767161530048,
1.33065236626948887767161530048, 2.51221127919791218134376952725, 3.55157884318988481737082954886, 4.92295768648648171557765776395, 6.73413114950597946664975841638, 7.58641610201656756020224939661, 8.219926020571777198935681583159, 8.778239798398161218146474908086, 9.498977342807506019120465663396, 10.26726801923786606480056160288
