| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s − 7-s + 4·10-s + 11-s − 2·14-s − 4·16-s − 2·17-s − 6·19-s + 4·20-s + 2·22-s + 2·23-s − 25-s − 2·28-s + 9·29-s + 9·31-s − 8·32-s − 4·34-s − 2·35-s − 4·37-s − 12·38-s − 3·41-s + 2·43-s + 2·44-s + 4·46-s + 12·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s − 0.377·7-s + 1.26·10-s + 0.301·11-s − 0.534·14-s − 16-s − 0.485·17-s − 1.37·19-s + 0.894·20-s + 0.426·22-s + 0.417·23-s − 1/5·25-s − 0.377·28-s + 1.67·29-s + 1.61·31-s − 1.41·32-s − 0.685·34-s − 0.338·35-s − 0.657·37-s − 1.94·38-s − 0.468·41-s + 0.304·43-s + 0.301·44-s + 0.589·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.76498715881568, −13.56794122605052, −12.88087522092889, −12.52668460618109, −12.06931768762332, −11.71058954875302, −10.88640270189699, −10.56354097955608, −10.05691732160955, −9.430243329811610, −8.925530343420814, −8.544780475297627, −7.854727343825173, −6.949723240840924, −6.633703958638142, −6.152576994621300, −5.893110580997889, −5.133253865185982, −4.598630905295481, −4.278067016906861, −3.607250511400229, −2.861325908254916, −2.527063587958897, −1.902333740332676, −1.037073850227847, 0,
1.037073850227847, 1.902333740332676, 2.527063587958897, 2.861325908254916, 3.607250511400229, 4.278067016906861, 4.598630905295481, 5.133253865185982, 5.893110580997889, 6.152576994621300, 6.633703958638142, 6.949723240840924, 7.854727343825173, 8.544780475297627, 8.925530343420814, 9.430243329811610, 10.05691732160955, 10.56354097955608, 10.88640270189699, 11.71058954875302, 12.06931768762332, 12.52668460618109, 12.88087522092889, 13.56794122605052, 13.76498715881568
