| L(s) = 1 | + 3·3-s − 5·5-s + 29.9·7-s + 9·9-s − 30.8·11-s + 13·13-s − 15·15-s + 25.6·17-s + 106.·19-s + 89.8·21-s + 31.6·23-s + 25·25-s + 27·27-s − 2.43·29-s − 46.5·31-s − 92.5·33-s − 149.·35-s + 45.9·37-s + 39·39-s − 281.·41-s + 13.2·43-s − 45·45-s − 60.9·47-s + 553.·49-s + 76.9·51-s + 292.·53-s + 154.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.61·7-s + 0.333·9-s − 0.845·11-s + 0.277·13-s − 0.258·15-s + 0.365·17-s + 1.29·19-s + 0.933·21-s + 0.286·23-s + 0.200·25-s + 0.192·27-s − 0.0155·29-s − 0.269·31-s − 0.488·33-s − 0.722·35-s + 0.204·37-s + 0.160·39-s − 1.07·41-s + 0.0468·43-s − 0.149·45-s − 0.189·47-s + 1.61·49-s + 0.211·51-s + 0.758·53-s + 0.378·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.250349953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.250349953\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 13 | \( 1 - 13T \) |
| good | 7 | \( 1 - 29.9T + 343T^{2} \) |
| 11 | \( 1 + 30.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 25.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43T + 2.43e4T^{2} \) |
| 31 | \( 1 + 46.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 45.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 281.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 13.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 60.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 372.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 314.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 25.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 34.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 544.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 993.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 528.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 927.T + 9.12e5T^{2} \) |
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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
−8.846719423958381472701693817603, −8.162350633185212610201425708749, −7.70762886872574348312822776032, −6.96792290391494825254353574199, −5.48155248539699662711163587888, −4.98658594598917044645159385573, −3.97879354278632255975080625296, −3.00370464873018555189127875768, −1.91403277182000521456956724027, −0.892899135186258587421945627216,
0.892899135186258587421945627216, 1.91403277182000521456956724027, 3.00370464873018555189127875768, 3.97879354278632255975080625296, 4.98658594598917044645159385573, 5.48155248539699662711163587888, 6.96792290391494825254353574199, 7.70762886872574348312822776032, 8.162350633185212610201425708749, 8.846719423958381472701693817603
