| L(s) = 1 | − 5-s + 4·7-s + 6·13-s + 3·17-s + 4·19-s + 23-s + 25-s − 8·29-s + 5·31-s − 4·35-s − 4·37-s − 2·41-s − 5·47-s + 9·49-s − 13·53-s + 8·59-s − 11·61-s − 6·65-s + 10·67-s − 6·71-s + 4·73-s + 5·79-s + 4·83-s − 3·85-s − 12·89-s + 24·91-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.66·13-s + 0.727·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.48·29-s + 0.898·31-s − 0.676·35-s − 0.657·37-s − 0.312·41-s − 0.729·47-s + 9/7·49-s − 1.78·53-s + 1.04·59-s − 1.40·61-s − 0.744·65-s + 1.22·67-s − 0.712·71-s + 0.468·73-s + 0.562·79-s + 0.439·83-s − 0.325·85-s − 1.27·89-s + 2.51·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.388676304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.388676304\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.61239537952978, −14.15708208772392, −13.79273264815407, −13.18942660071874, −12.59720976467712, −11.94212391662280, −11.42789927808009, −11.19739250527568, −10.70689151401868, −10.00866213486594, −9.365093861805828, −8.713609345749764, −8.278857816802699, −7.811454814605926, −7.402772590138352, −6.604561058684720, −5.955062214422765, −5.337415646483482, −4.882472183719536, −4.184263290762184, −3.518761368565563, −3.088971045653812, −1.881243646928165, −1.446313140224272, −0.7158511945983851,
0.7158511945983851, 1.446313140224272, 1.881243646928165, 3.088971045653812, 3.518761368565563, 4.184263290762184, 4.882472183719536, 5.337415646483482, 5.955062214422765, 6.604561058684720, 7.402772590138352, 7.811454814605926, 8.278857816802699, 8.713609345749764, 9.365093861805828, 10.00866213486594, 10.70689151401868, 11.19739250527568, 11.42789927808009, 11.94212391662280, 12.59720976467712, 13.18942660071874, 13.79273264815407, 14.15708208772392, 14.61239537952978
