| L(s) = 1 | + 2·2-s + 4·4-s − 12·7-s + 8·8-s + 9·9-s − 4·11-s − 13·13-s − 24·14-s + 16·16-s − 18·17-s + 18·18-s + 12·19-s − 8·22-s + 25·25-s − 26·26-s − 48·28-s + 6·29-s + 36·31-s + 32·32-s − 36·34-s + 36·36-s + 24·38-s − 16·44-s + 68·47-s + 95·49-s + 50·50-s − 52·52-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 1.71·7-s + 8-s + 9-s − 0.363·11-s − 13-s − 1.71·14-s + 16-s − 1.05·17-s + 18-s + 0.631·19-s − 0.363·22-s + 25-s − 26-s − 1.71·28-s + 6/29·29-s + 1.16·31-s + 32-s − 1.05·34-s + 36-s + 0.631·38-s − 0.363·44-s + 1.44·47-s + 1.93·49-s + 50-s − 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.778329854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.778329854\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 12 T + p^{2} T^{2} \) |
| 11 | \( 1 + 4 T + p^{2} T^{2} \) |
| 17 | \( 1 + 18 T + p^{2} T^{2} \) |
| 19 | \( 1 - 12 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 6 T + p^{2} T^{2} \) |
| 31 | \( 1 - 36 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 68 T + p^{2} T^{2} \) |
| 53 | \( 1 + 102 T + p^{2} T^{2} \) |
| 59 | \( 1 + 116 T + p^{2} T^{2} \) |
| 61 | \( 1 + 86 T + p^{2} T^{2} \) |
| 67 | \( 1 - 108 T + p^{2} T^{2} \) |
| 71 | \( 1 + 92 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 + 68 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−15.42804640739273657912802023574, −13.85515231997384645134636816410, −12.90283936863142454027735835960, −12.27118507569251615933155810753, −10.57460291292869959367231467685, −9.558707029092064347361325639130, −7.30253010602254279549104896775, −6.36235959030454351930595830061, −4.60023494735922761720874208202, −2.91906282025401256302203570916,
2.91906282025401256302203570916, 4.60023494735922761720874208202, 6.36235959030454351930595830061, 7.30253010602254279549104896775, 9.558707029092064347361325639130, 10.57460291292869959367231467685, 12.27118507569251615933155810753, 12.90283936863142454027735835960, 13.85515231997384645134636816410, 15.42804640739273657912802023574
