| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 13-s + 15-s − 2·17-s − 4·21-s + 25-s + 27-s + 2·29-s − 4·31-s − 4·35-s + 6·37-s − 39-s + 6·41-s − 4·43-s + 45-s − 4·47-s + 9·49-s − 2·51-s − 10·53-s + 2·61-s − 4·63-s − 65-s + 8·67-s + 4·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.485·17-s − 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.986·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s + 0.256·61-s − 0.503·63-s − 0.124·65-s + 0.977·67-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.38335805168842, −12.82468670655741, −12.61388533093771, −12.19963606092073, −11.28884731507220, −11.08160398407698, −10.36459313867703, −9.859328875904667, −9.597547865184474, −9.211483208932268, −8.728225473533760, −8.065076794890481, −7.672394625225608, −6.969532199270895, −6.516925164349866, −6.329368803675812, −5.530030941271884, −5.110606727260080, −4.285027021893807, −3.923688458246933, −3.183956638250900, −2.850785751453231, −2.282409828701679, −1.632239815936179, −0.7675821797723314, 0,
0.7675821797723314, 1.632239815936179, 2.282409828701679, 2.850785751453231, 3.183956638250900, 3.923688458246933, 4.285027021893807, 5.110606727260080, 5.530030941271884, 6.329368803675812, 6.516925164349866, 6.969532199270895, 7.672394625225608, 8.065076794890481, 8.728225473533760, 9.211483208932268, 9.597547865184474, 9.859328875904667, 10.36459313867703, 11.08160398407698, 11.28884731507220, 12.19963606092073, 12.61388533093771, 12.82468670655741, 13.38335805168842
