| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 7-s + 3·8-s − 2·9-s − 6·11-s + 12-s + 6·13-s − 14-s − 16-s + 2·18-s − 5·19-s − 21-s + 6·22-s − 3·24-s − 6·26-s + 5·27-s − 28-s − 9·29-s − 7·31-s − 5·32-s + 6·33-s + 2·36-s − 10·37-s + 5·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 0.377·7-s + 1.06·8-s − 2/3·9-s − 1.80·11-s + 0.288·12-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.471·18-s − 1.14·19-s − 0.218·21-s + 1.27·22-s − 0.612·24-s − 1.17·26-s + 0.962·27-s − 0.188·28-s − 1.67·29-s − 1.25·31-s − 0.883·32-s + 1.04·33-s + 1/3·36-s − 1.64·37-s + 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.90523195565883, −14.08577742847873, −13.71345580737472, −13.19643443616035, −12.79645937777070, −12.29022412788483, −11.33352551165969, −10.98038476654013, −10.68894453239998, −10.34621571391099, −9.487645692181553, −8.874925587308508, −8.551702973257487, −8.055815845636921, −7.593674392406642, −6.903617348060767, −6.073763038292061, −5.629007604444993, −5.130463222928472, −4.641677684018526, −3.696683373806745, −3.339233010511524, −2.141497508733951, −1.724436647697087, −0.6084386670055603, 0,
0.6084386670055603, 1.724436647697087, 2.141497508733951, 3.339233010511524, 3.696683373806745, 4.641677684018526, 5.130463222928472, 5.629007604444993, 6.073763038292061, 6.903617348060767, 7.593674392406642, 8.055815845636921, 8.551702973257487, 8.874925587308508, 9.487645692181553, 10.34621571391099, 10.68894453239998, 10.98038476654013, 11.33352551165969, 12.29022412788483, 12.79645937777070, 13.19643443616035, 13.71345580737472, 14.08577742847873, 14.90523195565883
