| L(s) = 1 | + 2.63·2-s + 4.93·4-s − 1.79·7-s + 7.73·8-s + 1.80·11-s + 1.97·13-s − 4.73·14-s + 10.4·16-s + 4.80·17-s + 2.96·19-s + 4.76·22-s − 1.73·23-s + 5.19·26-s − 8.87·28-s − 7.36·29-s − 2.62·31-s + 12.1·32-s + 12.6·34-s + 11.6·37-s + 7.80·38-s − 2.46·41-s − 7.27·43-s + 8.92·44-s − 4.56·46-s + 6.29·47-s − 3.76·49-s + 9.73·52-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.46·4-s − 0.679·7-s + 2.73·8-s + 0.545·11-s + 0.546·13-s − 1.26·14-s + 2.62·16-s + 1.16·17-s + 0.680·19-s + 1.01·22-s − 0.361·23-s + 1.01·26-s − 1.67·28-s − 1.36·29-s − 0.471·31-s + 2.15·32-s + 2.17·34-s + 1.91·37-s + 1.26·38-s − 0.385·41-s − 1.10·43-s + 1.34·44-s − 0.673·46-s + 0.917·47-s − 0.538·49-s + 1.34·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.843611427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.843611427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 7.27T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 9.10T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 - 1.09T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 3.83T + 97T^{2} \) |
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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
−9.333610620467687275902244778082, −8.014214724991627466486144849396, −7.25989779213691112293274243527, −6.46906113105488149454462881169, −5.80541952820827538846547857482, −5.21219836113836705714892453675, −4.03444666176582085483552494455, −3.56560251152700243403749272035, −2.70638916567970191410781593872, −1.43521928269557906204356746021,
1.43521928269557906204356746021, 2.70638916567970191410781593872, 3.56560251152700243403749272035, 4.03444666176582085483552494455, 5.21219836113836705714892453675, 5.80541952820827538846547857482, 6.46906113105488149454462881169, 7.25989779213691112293274243527, 8.014214724991627466486144849396, 9.333610620467687275902244778082
