| L(s) = 1 | + 2·3-s − 5-s + 2·7-s + 9-s + 4·11-s − 13-s − 2·15-s + 2·17-s + 4·21-s − 6·23-s + 25-s − 4·27-s − 10·29-s + 8·33-s − 2·35-s + 10·37-s − 2·39-s − 2·41-s + 2·43-s − 45-s − 6·47-s − 3·49-s + 4·51-s + 2·53-s − 4·55-s − 8·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.516·15-s + 0.485·17-s + 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.85·29-s + 1.39·33-s − 0.338·35-s + 1.64·37-s − 0.320·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.539·55-s − 1.04·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.764419164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764419164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−11.86211802977140750906415972760, −11.24644254703426645988179324297, −9.804097713553654747996703799713, −9.029007054104177535540216852388, −8.075719677783827392213832045836, −7.44804443925374277002602322353, −5.95797413547964079007979268212, −4.38508404435448495799335874607, −3.41867360196891109068838701089, −1.87254010620899892030103745588,
1.87254010620899892030103745588, 3.41867360196891109068838701089, 4.38508404435448495799335874607, 5.95797413547964079007979268212, 7.44804443925374277002602322353, 8.075719677783827392213832045836, 9.029007054104177535540216852388, 9.804097713553654747996703799713, 11.24644254703426645988179324297, 11.86211802977140750906415972760
