| L(s) = 1 | + 2-s + 4-s + 8-s + 11-s − 4·13-s + 16-s − 17-s + 3·19-s + 22-s + 23-s − 5·25-s − 4·26-s + 29-s − 6·31-s + 32-s − 34-s − 3·37-s + 3·38-s − 6·41-s + 43-s + 44-s + 46-s − 47-s − 5·50-s − 4·52-s + 58-s + 7·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s − 1.10·13-s + 1/4·16-s − 0.242·17-s + 0.688·19-s + 0.213·22-s + 0.208·23-s − 25-s − 0.784·26-s + 0.185·29-s − 1.07·31-s + 0.176·32-s − 0.171·34-s − 0.493·37-s + 0.486·38-s − 0.937·41-s + 0.152·43-s + 0.150·44-s + 0.147·46-s − 0.145·47-s − 0.707·50-s − 0.554·52-s + 0.131·58-s + 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.17771510177222704790130645353, −6.74491068671730722268168079203, −5.78264411571812896829987777620, −5.30390016817710142270316814947, −4.57756055657851239546035527260, −3.85299022907865807887494069853, −3.11080416218312545614960133256, −2.28545100223998648852991739349, −1.45211089439733810757236744771, 0,
1.45211089439733810757236744771, 2.28545100223998648852991739349, 3.11080416218312545614960133256, 3.85299022907865807887494069853, 4.57756055657851239546035527260, 5.30390016817710142270316814947, 5.78264411571812896829987777620, 6.74491068671730722268168079203, 7.17771510177222704790130645353
