| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s − 11-s − 14-s + 16-s − 2·17-s + 3·18-s − 3·19-s + 22-s + 3·23-s − 5·25-s + 28-s − 3·29-s − 4·31-s − 32-s + 2·34-s − 3·36-s + 2·37-s + 3·38-s − 6·41-s + 43-s − 44-s − 3·46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s − 0.301·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.688·19-s + 0.213·22-s + 0.625·23-s − 25-s + 0.188·28-s − 0.557·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.328·37-s + 0.486·38-s − 0.937·41-s + 0.152·43-s − 0.150·44-s − 0.442·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5568526867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5568526867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.20866630086175, −15.07154850940525, −14.18495089891640, −13.88657143467977, −13.11387901170578, −12.60092745115072, −11.95115072717202, −11.34895354939486, −10.96239716243403, −10.60383216046358, −9.685175375510959, −9.312946164537083, −8.687908495034666, −8.146564053731911, −7.769055970530390, −6.987528837606725, −6.404688205807572, −5.755908045378048, −5.211161216904139, −4.449674387839522, −3.609772423985277, −2.905119607233705, −2.169260431156132, −1.544219735352382, −0.3174261047079639,
0.3174261047079639, 1.544219735352382, 2.169260431156132, 2.905119607233705, 3.609772423985277, 4.449674387839522, 5.211161216904139, 5.755908045378048, 6.404688205807572, 6.987528837606725, 7.769055970530390, 8.146564053731911, 8.687908495034666, 9.312946164537083, 9.685175375510959, 10.60383216046358, 10.96239716243403, 11.34895354939486, 11.95115072717202, 12.60092745115072, 13.11387901170578, 13.88657143467977, 14.18495089891640, 15.07154850940525, 15.20866630086175
