| L(s) = 1 | + 3-s − 2·5-s − 7-s − 2·9-s + 11-s − 3·13-s − 2·15-s + 4·17-s − 2·19-s − 21-s + 9·23-s − 25-s − 5·27-s + 29-s − 7·31-s + 33-s + 2·35-s + 4·37-s − 3·39-s − 12·41-s + 3·43-s + 4·45-s + 49-s + 4·51-s + 3·53-s − 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.832·13-s − 0.516·15-s + 0.970·17-s − 0.458·19-s − 0.218·21-s + 1.87·23-s − 1/5·25-s − 0.962·27-s + 0.185·29-s − 1.25·31-s + 0.174·33-s + 0.338·35-s + 0.657·37-s − 0.480·39-s − 1.87·41-s + 0.457·43-s + 0.596·45-s + 1/7·49-s + 0.560·51-s + 0.412·53-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.56105942393439, −13.24655195844077, −12.57290407517906, −12.18490167885985, −11.81641755102760, −11.27510634729869, −10.76400564854342, −10.34883688391421, −9.580511152078067, −9.211207314246536, −8.872907423403437, −8.241820540730147, −7.725722361881214, −7.417585438730196, −6.898531586105930, −6.217738987663191, −5.704706685321645, −4.997228606531266, −4.647174343828485, −3.836276335374943, −3.241496573777286, −3.156519968241949, −2.311977931024148, −1.635040468663183, −0.6999870570543671, 0,
0.6999870570543671, 1.635040468663183, 2.311977931024148, 3.156519968241949, 3.241496573777286, 3.836276335374943, 4.647174343828485, 4.997228606531266, 5.704706685321645, 6.217738987663191, 6.898531586105930, 7.417585438730196, 7.725722361881214, 8.241820540730147, 8.872907423403437, 9.211207314246536, 9.580511152078067, 10.34883688391421, 10.76400564854342, 11.27510634729869, 11.81641755102760, 12.18490167885985, 12.57290407517906, 13.24655195844077, 13.56105942393439
