| L(s) = 1 | − 3-s − 2.15·5-s − 0.0890·7-s + 9-s + 6.37·11-s + 3.74·13-s + 2.15·15-s + 4.37·17-s + 5.14·19-s + 0.0890·21-s + 1.37·23-s − 0.342·25-s − 27-s + 3.68·29-s + 10.8·31-s − 6.37·33-s + 0.192·35-s + 2.87·37-s − 3.74·39-s + 2.86·41-s − 8.06·43-s − 2.15·45-s − 7.11·47-s − 6.99·49-s − 4.37·51-s + 1.35·53-s − 13.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.965·5-s − 0.0336·7-s + 0.333·9-s + 1.92·11-s + 1.03·13-s + 0.557·15-s + 1.06·17-s + 1.18·19-s + 0.0194·21-s + 0.285·23-s − 0.0685·25-s − 0.192·27-s + 0.685·29-s + 1.94·31-s − 1.10·33-s + 0.0324·35-s + 0.472·37-s − 0.599·39-s + 0.448·41-s − 1.22·43-s − 0.321·45-s − 1.03·47-s − 0.998·49-s − 0.612·51-s + 0.185·53-s − 1.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877477725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877477725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 2.15T + 5T^{2} \) |
| 7 | \( 1 + 0.0890T + 7T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 5.14T + 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 - 3.68T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 2.87T + 37T^{2} \) |
| 41 | \( 1 - 2.86T + 41T^{2} \) |
| 43 | \( 1 + 8.06T + 43T^{2} \) |
| 47 | \( 1 + 7.11T + 47T^{2} \) |
| 53 | \( 1 - 1.35T + 53T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 - 1.63T + 61T^{2} \) |
| 67 | \( 1 - 6.57T + 67T^{2} \) |
| 71 | \( 1 - 5.51T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 0.145T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 3.70T + 89T^{2} \) |
| 97 | \( 1 - 8.43T + 97T^{2} \) |
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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
−8.111456658537639820372209784268, −7.33197860549310194196208673064, −6.50458343469031013980902303367, −6.19201167440018701038866526371, −5.14652074768856979188633895751, −4.37168956475097823665189642739, −3.67284553756834003310304244212, −3.12643283846225160926561597103, −1.40041538008891924785107821457, −0.872869417502735893112153580145,
0.872869417502735893112153580145, 1.40041538008891924785107821457, 3.12643283846225160926561597103, 3.67284553756834003310304244212, 4.37168956475097823665189642739, 5.14652074768856979188633895751, 6.19201167440018701038866526371, 6.50458343469031013980902303367, 7.33197860549310194196208673064, 8.111456658537639820372209784268
