| L(s) = 1 | + 2.95·3-s + 2.29·5-s + 2.82·7-s + 5.73·9-s + 0.493·11-s − 1.57·13-s + 6.76·15-s − 3.39·17-s + 1.17·19-s + 8.35·21-s − 0.0257·23-s + 0.244·25-s + 8.09·27-s − 4.46·29-s + 6.43·31-s + 1.45·33-s + 6.47·35-s − 5.77·37-s − 4.66·39-s + 5.69·41-s + 7.57·43-s + 13.1·45-s − 3.16·47-s + 0.990·49-s − 10.0·51-s + 5.63·53-s + 1.13·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s + 1.02·5-s + 1.06·7-s + 1.91·9-s + 0.148·11-s − 0.437·13-s + 1.74·15-s − 0.824·17-s + 0.268·19-s + 1.82·21-s − 0.00537·23-s + 0.0488·25-s + 1.55·27-s − 0.829·29-s + 1.15·31-s + 0.254·33-s + 1.09·35-s − 0.949·37-s − 0.746·39-s + 0.889·41-s + 1.15·43-s + 1.95·45-s − 0.461·47-s + 0.141·49-s − 1.40·51-s + 0.774·53-s + 0.152·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.031993641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.031993641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.95T + 3T^{2} \) |
| 5 | \( 1 - 2.29T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.493T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 0.0257T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 + 5.77T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 - 5.63T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 - 1.95T + 71T^{2} \) |
| 73 | \( 1 - 0.678T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 + 2.38T + 83T^{2} \) |
| 89 | \( 1 + 0.754T + 89T^{2} \) |
| 97 | \( 1 - 0.821T + 97T^{2} \) |
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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
−8.584149325572100182657543677444, −7.79023623006836266855627162215, −7.27650312311647460998338578383, −6.33577466541973549374815663577, −5.35502352596971359321044138063, −4.53896156386559849462076641781, −3.78599188772714976845811779123, −2.65617636031311290401628564420, −2.15121442147826592026378440303, −1.37711002573201581326433405018,
1.37711002573201581326433405018, 2.15121442147826592026378440303, 2.65617636031311290401628564420, 3.78599188772714976845811779123, 4.53896156386559849462076641781, 5.35502352596971359321044138063, 6.33577466541973549374815663577, 7.27650312311647460998338578383, 7.79023623006836266855627162215, 8.584149325572100182657543677444
