| L(s) = 1 | + 5-s − 2·7-s − 3·13-s − 4·17-s + 4·19-s − 23-s + 25-s − 29-s − 31-s − 2·35-s − 8·37-s − 11·41-s + 10·43-s − 47-s − 3·49-s + 6·53-s − 8·59-s − 8·61-s − 3·65-s − 12·67-s + 13·71-s + 7·73-s + 12·79-s + 16·83-s − 4·85-s + 6·89-s + 6·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.832·13-s − 0.970·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.185·29-s − 0.179·31-s − 0.338·35-s − 1.31·37-s − 1.71·41-s + 1.52·43-s − 0.145·47-s − 3/7·49-s + 0.824·53-s − 1.04·59-s − 1.02·61-s − 0.372·65-s − 1.46·67-s + 1.54·71-s + 0.819·73-s + 1.35·79-s + 1.75·83-s − 0.433·85-s + 0.635·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.378143654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378143654\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.82231491862514, −15.44486584510704, −14.89906538665971, −14.05886160025830, −13.78052509623205, −13.19925391548905, −12.56932527538557, −12.09484892120511, −11.54513385415584, −10.67352492988755, −10.33773933147236, −9.601041898148794, −9.202647283375505, −8.682337380453179, −7.703586979703443, −7.296335001364823, −6.491543016620165, −6.167830945196884, −5.143352803839331, −4.888825899033368, −3.801365771871034, −3.224480409139656, −2.399485881293459, −1.723070700423963, −0.4835592090710512,
0.4835592090710512, 1.723070700423963, 2.399485881293459, 3.224480409139656, 3.801365771871034, 4.888825899033368, 5.143352803839331, 6.167830945196884, 6.491543016620165, 7.296335001364823, 7.703586979703443, 8.682337380453179, 9.202647283375505, 9.601041898148794, 10.33773933147236, 10.67352492988755, 11.54513385415584, 12.09484892120511, 12.56932527538557, 13.19925391548905, 13.78052509623205, 14.05886160025830, 14.89906538665971, 15.44486584510704, 15.82231491862514
