L(s) = 1 | + 2·2-s + 4·4-s + 6·5-s + 7·7-s + 8·8-s + 12·10-s + 30·11-s + 2·13-s + 14·14-s + 16·16-s + 66·17-s − 52·19-s + 24·20-s + 60·22-s + 114·23-s − 89·25-s + 4·26-s + 28·28-s + 72·29-s − 196·31-s + 32·32-s + 132·34-s + 42·35-s − 286·37-s − 104·38-s + 48·40-s − 378·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.536·5-s + 0.377·7-s + 0.353·8-s + 0.379·10-s + 0.822·11-s + 0.0426·13-s + 0.267·14-s + 1/4·16-s + 0.941·17-s − 0.627·19-s + 0.268·20-s + 0.581·22-s + 1.03·23-s − 0.711·25-s + 0.0301·26-s + 0.188·28-s + 0.461·29-s − 1.13·31-s + 0.176·32-s + 0.665·34-s + 0.202·35-s − 1.27·37-s − 0.443·38-s + 0.189·40-s − 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.815812138\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.815812138\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 72 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 228 T + p^{3} T^{2} \) |
| 53 | \( 1 + 348 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 106 T + p^{3} T^{2} \) |
| 67 | \( 1 - 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 630 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1042 T + p^{3} T^{2} \) |
| 79 | \( 1 + 88 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1440 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1374 T + p^{3} T^{2} \) |
| 97 | \( 1 + 34 T + p^{3} T^{2} \) |
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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
−12.94091499488990099968496348899, −12.00469864117932881928923059867, −11.00283492915201850036700851524, −9.878234365874890508013735505997, −8.642730572384925408026385328640, −7.22420322337839517580687188879, −6.07037266119863582013056580460, −4.93036297379422592879321219661, −3.47092576784618890207366984219, −1.68755606271270684431715349395,
1.68755606271270684431715349395, 3.47092576784618890207366984219, 4.93036297379422592879321219661, 6.07037266119863582013056580460, 7.22420322337839517580687188879, 8.642730572384925408026385328640, 9.878234365874890508013735505997, 11.00283492915201850036700851524, 12.00469864117932881928923059867, 12.94091499488990099968496348899