| L(s) = 1 | + 3-s − 2·5-s − 4·7-s − 2·9-s − 2·11-s − 7·13-s − 2·15-s − 4·17-s − 4·21-s − 23-s − 25-s − 5·27-s − 5·29-s − 3·31-s − 2·33-s + 8·35-s − 2·37-s − 7·39-s + 9·41-s + 8·43-s + 4·45-s − 47-s + 9·49-s − 4·51-s + 6·53-s + 4·55-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s − 2/3·9-s − 0.603·11-s − 1.94·13-s − 0.516·15-s − 0.970·17-s − 0.872·21-s − 0.208·23-s − 1/5·25-s − 0.962·27-s − 0.928·29-s − 0.538·31-s − 0.348·33-s + 1.35·35-s − 0.328·37-s − 1.12·39-s + 1.40·41-s + 1.21·43-s + 0.596·45-s − 0.145·47-s + 9/7·49-s − 0.560·51-s + 0.824·53-s + 0.539·55-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.53417647104844, −13.97701611454550, −13.35085723116462, −12.97205699095989, −12.40514858907994, −12.12939228606258, −11.44468716022179, −10.92471863473233, −10.38656773128605, −9.608657202972680, −9.452078921537891, −8.933983897387215, −8.222154752185513, −7.687204952988552, −7.269637313129846, −6.875389318207914, −6.003292733794517, −5.595684397432836, −4.861726399184781, −4.106785349525145, −3.727608619298274, −2.955797576723941, −2.543681852928984, −2.084017715991093, −0.4698655587319938, 0,
0.4698655587319938, 2.084017715991093, 2.543681852928984, 2.955797576723941, 3.727608619298274, 4.106785349525145, 4.861726399184781, 5.595684397432836, 6.003292733794517, 6.875389318207914, 7.269637313129846, 7.687204952988552, 8.222154752185513, 8.933983897387215, 9.452078921537891, 9.608657202972680, 10.38656773128605, 10.92471863473233, 11.44468716022179, 12.12939228606258, 12.40514858907994, 12.97205699095989, 13.35085723116462, 13.97701611454550, 14.53417647104844
