| L(s) = 1 | − 3-s − 5-s − 2·9-s + 2·11-s + 2·13-s + 15-s + 2·17-s + 23-s + 25-s + 5·27-s + 29-s + 2·31-s − 2·33-s − 4·37-s − 2·39-s − 5·41-s − 11·43-s + 2·45-s − 8·47-s − 2·51-s + 8·53-s − 2·55-s − 4·59-s − 5·61-s − 2·65-s + 67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.603·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 0.185·29-s + 0.359·31-s − 0.348·33-s − 0.657·37-s − 0.320·39-s − 0.780·41-s − 1.67·43-s + 0.298·45-s − 1.16·47-s − 0.280·51-s + 1.09·53-s − 0.269·55-s − 0.520·59-s − 0.640·61-s − 0.248·65-s + 0.122·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−16.44297012752525, −15.76353030393401, −15.15360065601586, −14.65215521129626, −14.10489766831191, −13.46443397321197, −12.95273986334730, −12.02353521787661, −11.83717111410630, −11.41822300664701, −10.57560696199566, −10.28942633616393, −9.389765366470202, −8.742695938164380, −8.321780804534997, −7.635697842251592, −6.783307270513412, −6.404311518633095, −5.685142314759815, −5.040056501642286, −4.426806998110103, −3.444324152990580, −3.129664384545378, −1.896817494136521, −1.009852983427789, 0,
1.009852983427789, 1.896817494136521, 3.129664384545378, 3.444324152990580, 4.426806998110103, 5.040056501642286, 5.685142314759815, 6.404311518633095, 6.783307270513412, 7.635697842251592, 8.321780804534997, 8.742695938164380, 9.389765366470202, 10.28942633616393, 10.57560696199566, 11.41822300664701, 11.83717111410630, 12.02353521787661, 12.95273986334730, 13.46443397321197, 14.10489766831191, 14.65215521129626, 15.15360065601586, 15.76353030393401, 16.44297012752525
