| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 3·11-s − 4·13-s + 4·14-s + 16-s − 4·19-s − 3·22-s + 4·26-s − 4·28-s + 6·29-s − 31-s − 32-s − 4·37-s + 4·38-s + 6·41-s + 5·43-s + 3·44-s + 12·47-s + 9·49-s − 4·52-s − 6·53-s + 4·56-s − 6·58-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 0.904·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.917·19-s − 0.639·22-s + 0.784·26-s − 0.755·28-s + 1.11·29-s − 0.179·31-s − 0.176·32-s − 0.657·37-s + 0.648·38-s + 0.937·41-s + 0.762·43-s + 0.452·44-s + 1.75·47-s + 9/7·49-s − 0.554·52-s − 0.824·53-s + 0.534·56-s − 0.787·58-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.62615610647280, −14.21059440050392, −13.61071855984741, −12.95246523999863, −12.41863022574844, −12.21001553607089, −11.64161380024133, −10.87209233443488, −10.26842466088033, −10.13903413148605, −9.333554031926662, −9.009336498694306, −8.722200284788252, −7.591986441607940, −7.459165208873890, −6.727338902643677, −6.186319744972871, −6.000028617206492, −4.954339524598431, −4.298197729268344, −3.703643602732639, −2.906375959401255, −2.539749958978054, −1.666218994923966, −0.7357042120715595, 0,
0.7357042120715595, 1.666218994923966, 2.539749958978054, 2.906375959401255, 3.703643602732639, 4.298197729268344, 4.954339524598431, 6.000028617206492, 6.186319744972871, 6.727338902643677, 7.459165208873890, 7.591986441607940, 8.722200284788252, 9.009336498694306, 9.333554031926662, 10.13903413148605, 10.26842466088033, 10.87209233443488, 11.64161380024133, 12.21001553607089, 12.41863022574844, 12.95246523999863, 13.61071855984741, 14.21059440050392, 14.62615610647280
