| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 12-s − 14-s − 15-s − 16-s + 17-s − 18-s − 20-s − 21-s − 2·23-s − 3·24-s + 25-s − 27-s − 28-s − 6·29-s + 30-s + 8·31-s − 5·32-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.417·23-s − 0.612·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s + 1.43·31-s − 0.883·32-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.91252431290356, −12.41371730565751, −11.85930383161187, −11.60811779958786, −10.86558230642303, −10.55177384352239, −10.17491555251436, −9.721314816077019, −9.262734235065586, −8.848317952350626, −8.298814935121479, −7.851961520260567, −7.497350779742807, −6.842775336579460, −6.346712812367794, −5.889523802360433, −5.229409951167709, −4.866199046900893, −4.539219983741231, −3.714350951790120, −3.367262732859509, −2.445139758975330, −1.721178006205982, −1.454577530970356, −0.6480145411779526, 0,
0.6480145411779526, 1.454577530970356, 1.721178006205982, 2.445139758975330, 3.367262732859509, 3.714350951790120, 4.539219983741231, 4.866199046900893, 5.229409951167709, 5.889523802360433, 6.346712812367794, 6.842775336579460, 7.497350779742807, 7.851961520260567, 8.298814935121479, 8.848317952350626, 9.262734235065586, 9.721314816077019, 10.17491555251436, 10.55177384352239, 10.86558230642303, 11.60811779958786, 11.85930383161187, 12.41371730565751, 12.91252431290356
