| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s − 4·11-s − 2·13-s + 16-s + 2·17-s − 3·18-s − 4·19-s − 4·22-s − 2·26-s − 6·29-s − 4·31-s + 32-s + 2·34-s − 3·36-s + 37-s − 4·38-s − 6·41-s − 4·43-s − 4·44-s + 8·47-s − 7·49-s − 2·52-s − 10·53-s − 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.852·22-s − 0.392·26-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.164·37-s − 0.648·38-s − 0.937·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s − 49-s − 0.277·52-s − 1.37·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.712726996024504289860030578877, −7.985100866972308578412355215355, −7.26997651195770461478664014699, −6.27355954369629806358703833439, −5.46325332967485333134031287798, −4.95477554784516684995777664760, −3.76639174593054485317551916841, −2.87123189491779009496507990789, −2.02250478001830636560137396847, 0,
2.02250478001830636560137396847, 2.87123189491779009496507990789, 3.76639174593054485317551916841, 4.95477554784516684995777664760, 5.46325332967485333134031287798, 6.27355954369629806358703833439, 7.26997651195770461478664014699, 7.985100866972308578412355215355, 8.712726996024504289860030578877
