| L(s) = 1 | + 7-s − 11-s + 2·13-s + 3·17-s + 7·19-s − 3·23-s − 6·29-s + 4·31-s + 5·37-s − 3·41-s + 4·43-s − 9·47-s − 6·49-s − 6·53-s − 3·59-s − 10·61-s + 10·67-s − 15·71-s + 2·73-s − 77-s + 79-s − 6·89-s + 2·91-s − 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.301·11-s + 0.554·13-s + 0.727·17-s + 1.60·19-s − 0.625·23-s − 1.11·29-s + 0.718·31-s + 0.821·37-s − 0.468·41-s + 0.609·43-s − 1.31·47-s − 6/7·49-s − 0.824·53-s − 0.390·59-s − 1.28·61-s + 1.22·67-s − 1.78·71-s + 0.234·73-s − 0.113·77-s + 0.112·79-s − 0.635·89-s + 0.209·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.93687795287023, −14.56731305778768, −13.99527508632272, −13.52466977787670, −13.09365764121995, −12.34193796243820, −11.97818453336701, −11.22950257649403, −11.12364509654398, −10.23602170760494, −9.717621283730092, −9.406144208405244, −8.586726565313532, −7.974831323734276, −7.689582058568478, −7.078712714250241, −6.203524703473583, −5.850393878300383, −5.125036207784399, −4.678105080896045, −3.804223209687243, −3.274142598385928, −2.640718541003530, −1.629129016430758, −1.149588449621270, 0,
1.149588449621270, 1.629129016430758, 2.640718541003530, 3.274142598385928, 3.804223209687243, 4.678105080896045, 5.125036207784399, 5.850393878300383, 6.203524703473583, 7.078712714250241, 7.689582058568478, 7.974831323734276, 8.586726565313532, 9.406144208405244, 9.717621283730092, 10.23602170760494, 11.12364509654398, 11.22950257649403, 11.97818453336701, 12.34193796243820, 13.09365764121995, 13.52466977787670, 13.99527508632272, 14.56731305778768, 14.93687795287023
