| L(s) = 1 | + 1.28·2-s + 3.30·3-s − 0.336·4-s + 4.25·6-s + 0.563·7-s − 3.01·8-s + 7.89·9-s + 11-s − 1.10·12-s − 1.69·13-s + 0.726·14-s − 3.21·16-s − 0.0949·17-s + 10.1·18-s + 19-s + 1.86·21-s + 1.28·22-s + 0.625·23-s − 9.94·24-s − 2.19·26-s + 16.1·27-s − 0.189·28-s + 6.14·29-s + 3.11·31-s + 1.88·32-s + 3.30·33-s − 0.122·34-s + ⋯ |
| L(s) = 1 | + 0.912·2-s + 1.90·3-s − 0.168·4-s + 1.73·6-s + 0.212·7-s − 1.06·8-s + 2.63·9-s + 0.301·11-s − 0.320·12-s − 0.471·13-s + 0.194·14-s − 0.803·16-s − 0.0230·17-s + 2.40·18-s + 0.229·19-s + 0.405·21-s + 0.275·22-s + 0.130·23-s − 2.03·24-s − 0.429·26-s + 3.10·27-s − 0.0358·28-s + 1.14·29-s + 0.559·31-s + 0.332·32-s + 0.574·33-s − 0.0210·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.083686557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.083686557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 - 0.563T + 7T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + 0.0949T + 17T^{2} \) |
| 23 | \( 1 - 0.625T + 23T^{2} \) |
| 29 | \( 1 - 6.14T + 29T^{2} \) |
| 31 | \( 1 - 3.11T + 31T^{2} \) |
| 37 | \( 1 - 2.91T + 37T^{2} \) |
| 41 | \( 1 - 0.562T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 8.90T + 47T^{2} \) |
| 53 | \( 1 + 0.489T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 2.05T + 67T^{2} \) |
| 71 | \( 1 - 3.06T + 71T^{2} \) |
| 73 | \( 1 + 5.61T + 73T^{2} \) |
| 79 | \( 1 + 5.37T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 6.13T + 97T^{2} \) |
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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
−8.290457101204523303021635981478, −7.59268145291499546728694673186, −6.87887437772439615281291649797, −6.00564064986472587900201730906, −4.91373634125869791728958088735, −4.33402098843267876097102487100, −3.74914553426473043209380193990, −2.86244839527890530189664005055, −2.44431525676286986894375510750, −1.14744459568178841947093311921,
1.14744459568178841947093311921, 2.44431525676286986894375510750, 2.86244839527890530189664005055, 3.74914553426473043209380193990, 4.33402098843267876097102487100, 4.91373634125869791728958088735, 6.00564064986472587900201730906, 6.87887437772439615281291649797, 7.59268145291499546728694673186, 8.290457101204523303021635981478
