| L(s) = 1 | + 2-s + 4-s − 1.16·5-s + 1.20·7-s + 8-s − 1.16·10-s + 6.54·11-s − 6.22·13-s + 1.20·14-s + 16-s + 5.41·17-s + 6.55·19-s − 1.16·20-s + 6.54·22-s − 5.20·23-s − 3.64·25-s − 6.22·26-s + 1.20·28-s + 7.94·29-s + 9.69·31-s + 32-s + 5.41·34-s − 1.39·35-s − 0.452·37-s + 6.55·38-s − 1.16·40-s − 3.44·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.519·5-s + 0.454·7-s + 0.353·8-s − 0.367·10-s + 1.97·11-s − 1.72·13-s + 0.321·14-s + 0.250·16-s + 1.31·17-s + 1.50·19-s − 0.259·20-s + 1.39·22-s − 1.08·23-s − 0.729·25-s − 1.22·26-s + 0.227·28-s + 1.47·29-s + 1.74·31-s + 0.176·32-s + 0.928·34-s − 0.236·35-s − 0.0743·37-s + 1.06·38-s − 0.183·40-s − 0.538·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.731277702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.731277702\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 1.16T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 - 6.54T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 - 9.69T + 31T^{2} \) |
| 37 | \( 1 + 0.452T + 37T^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 7.01T + 47T^{2} \) |
| 53 | \( 1 + 7.59T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 - 0.0143T + 73T^{2} \) |
| 79 | \( 1 - 6.36T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 0.749T + 89T^{2} \) |
| 97 | \( 1 + 4.71T + 97T^{2} \) |
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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
−7.72717107981244318101661151293, −7.13338502678484824885109456570, −6.41557929055418297849365266884, −5.71982698483265822076238291766, −4.80698103245323003454258524868, −4.42500092307362060107908874785, −3.53530335779122324967922274398, −2.92831121792697807788960078061, −1.78078645425863912274162357695, −0.909604410521646968975471649047,
0.909604410521646968975471649047, 1.78078645425863912274162357695, 2.92831121792697807788960078061, 3.53530335779122324967922274398, 4.42500092307362060107908874785, 4.80698103245323003454258524868, 5.71982698483265822076238291766, 6.41557929055418297849365266884, 7.13338502678484824885109456570, 7.72717107981244318101661151293
