| L(s) = 1 | + 2.88·2-s + 0.299·4-s + 5·5-s − 22.1·8-s + 14.4·10-s + 46.4·11-s − 31.0·13-s − 66.3·16-s + 61.8·17-s + 24.6·19-s + 1.49·20-s + 133.·22-s + 154.·23-s + 25·25-s − 89.4·26-s − 200.·29-s + 129.·31-s − 13.5·32-s + 178.·34-s − 77.9·37-s + 70.9·38-s − 110.·40-s − 235.·41-s − 278.·43-s + 13.9·44-s + 446.·46-s − 368.·47-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.0374·4-s + 0.447·5-s − 0.980·8-s + 0.455·10-s + 1.27·11-s − 0.662·13-s − 1.03·16-s + 0.882·17-s + 0.297·19-s + 0.0167·20-s + 1.29·22-s + 1.40·23-s + 0.200·25-s − 0.674·26-s − 1.28·29-s + 0.748·31-s − 0.0748·32-s + 0.898·34-s − 0.346·37-s + 0.302·38-s − 0.438·40-s − 0.896·41-s − 0.987·43-s + 0.0476·44-s + 1.43·46-s − 1.14·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.868286194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.868286194\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.88T + 8T^{2} \) |
| 11 | \( 1 - 46.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 77.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 169.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 691.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 696.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 2.33T + 3.00e5T^{2} \) |
| 71 | \( 1 - 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 752.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 842.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 23.4T + 9.12e5T^{2} \) |
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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
−8.894898602469773474242042167150, −7.85511235655180023320292182368, −6.79533558524005218845264003193, −6.31772376190970980001380987595, −5.21464986190638725240039852508, −4.93238017173652856929647828237, −3.71231190918797814094073408801, −3.21888289800798435304563618631, −1.97525490437392965992526973694, −0.77289365785794253476888403200,
0.77289365785794253476888403200, 1.97525490437392965992526973694, 3.21888289800798435304563618631, 3.71231190918797814094073408801, 4.93238017173652856929647828237, 5.21464986190638725240039852508, 6.31772376190970980001380987595, 6.79533558524005218845264003193, 7.85511235655180023320292182368, 8.894898602469773474242042167150
