| L(s) = 1 | + 3-s − 7-s + 9-s + 3.70·11-s + 2·13-s + 5.40·17-s + 7.70·19-s − 21-s − 23-s − 5·25-s + 27-s + 2·29-s − 3.40·31-s + 3.70·33-s + 3.40·37-s + 2·39-s − 3.70·41-s + 1.40·43-s + 6.29·47-s + 49-s + 5.40·51-s − 9.10·53-s + 7.70·57-s − 5.10·59-s + 6.29·61-s − 63-s + 13.4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 0.333·9-s + 1.11·11-s + 0.554·13-s + 1.31·17-s + 1.76·19-s − 0.218·21-s − 0.208·23-s − 25-s + 0.192·27-s + 0.371·29-s − 0.611·31-s + 0.644·33-s + 0.559·37-s + 0.320·39-s − 0.578·41-s + 0.213·43-s + 0.918·47-s + 0.142·49-s + 0.756·51-s − 1.25·53-s + 1.02·57-s − 0.664·59-s + 0.806·61-s − 0.125·63-s + 1.63·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.115520383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.115520383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.40T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 - 3.40T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 9.10T + 53T^{2} \) |
| 59 | \( 1 + 5.10T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 5.40T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.76955891031587878766423839136, −7.34874015662898059626013642731, −6.47621867360384587092488930658, −5.82871814383236284895405728985, −5.12341442530325497758611591396, −4.00238437579212229204889708246, −3.55452307930884782522302475185, −2.87293166624421301343379082092, −1.67372604256424874501373601747, −0.928812032894187998649617790708,
0.928812032894187998649617790708, 1.67372604256424874501373601747, 2.87293166624421301343379082092, 3.55452307930884782522302475185, 4.00238437579212229204889708246, 5.12341442530325497758611591396, 5.82871814383236284895405728985, 6.47621867360384587092488930658, 7.34874015662898059626013642731, 7.76955891031587878766423839136
