| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 9-s + 11-s − 14-s + 16-s + 18-s − 22-s + 25-s + 28-s − 32-s − 36-s − 2·37-s − 2·43-s + 44-s + 49-s − 50-s − 2·53-s − 56-s − 63-s + 64-s + 72-s + 2·74-s + 77-s − 2·79-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 9-s + 11-s − 14-s + 16-s + 18-s − 22-s + 25-s + 28-s − 32-s − 36-s − 2·37-s − 2·43-s + 44-s + 49-s − 50-s − 2·53-s − 56-s − 63-s + 64-s + 72-s + 2·74-s + 77-s − 2·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5606699344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5606699344\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−11.59508229068300331389283906517, −11.06949408754802463573739071329, −10.03361942961181878560731934145, −8.817683577267440290711413410854, −8.457616120483445713229058220927, −7.25990447407810617158114688362, −6.26303092825416067817048066405, −5.03389402404637893347715683091, −3.26460966876935290618034281553, −1.68088718624522364020573535673,
1.68088718624522364020573535673, 3.26460966876935290618034281553, 5.03389402404637893347715683091, 6.26303092825416067817048066405, 7.25990447407810617158114688362, 8.457616120483445713229058220927, 8.817683577267440290711413410854, 10.03361942961181878560731934145, 11.06949408754802463573739071329, 11.59508229068300331389283906517
