| L(s) = 1 | + 3.72·3-s − 5·5-s − 11.8·7-s − 13.1·9-s − 64.0·11-s − 77.5·13-s − 18.6·15-s + 25.6·17-s − 51.4·19-s − 44.2·21-s + 23·23-s + 25·25-s − 149.·27-s + 68.3·29-s + 67.5·31-s − 238.·33-s + 59.3·35-s − 149.·37-s − 289.·39-s − 249.·41-s + 534.·43-s + 65.5·45-s − 141.·47-s − 202.·49-s + 95.5·51-s − 76.1·53-s + 320.·55-s + ⋯ |
| L(s) = 1 | + 0.717·3-s − 0.447·5-s − 0.640·7-s − 0.485·9-s − 1.75·11-s − 1.65·13-s − 0.320·15-s + 0.365·17-s − 0.621·19-s − 0.459·21-s + 0.208·23-s + 0.200·25-s − 1.06·27-s + 0.437·29-s + 0.391·31-s − 1.25·33-s + 0.286·35-s − 0.663·37-s − 1.18·39-s − 0.951·41-s + 1.89·43-s + 0.217·45-s − 0.438·47-s − 0.589·49-s + 0.262·51-s − 0.197·53-s + 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7787849984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7787849984\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 3.72T + 27T^{2} \) |
| 7 | \( 1 + 11.8T + 343T^{2} \) |
| 11 | \( 1 + 64.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 77.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 68.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 534.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 76.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 816.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 356.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 746.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 162.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 194.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 246.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 836.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 53.5T + 9.12e5T^{2} \) |
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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
−8.746563136494355987553778269494, −8.082146647429873462742180433946, −7.51680820686752568841645671838, −6.69412032423101300365330604545, −5.48938155405196952396103773580, −4.89676075376555622008351438618, −3.70011865183593209073662464628, −2.74369280100729610905944360530, −2.35862327926274070842402398749, −0.36469104423109542783180348670,
0.36469104423109542783180348670, 2.35862327926274070842402398749, 2.74369280100729610905944360530, 3.70011865183593209073662464628, 4.89676075376555622008351438618, 5.48938155405196952396103773580, 6.69412032423101300365330604545, 7.51680820686752568841645671838, 8.082146647429873462742180433946, 8.746563136494355987553778269494
