| L(s) = 1 | − 4.54·2-s + 0.739·3-s + 12.6·4-s + 5.27·5-s − 3.36·6-s − 32.8·7-s − 21.0·8-s − 26.4·9-s − 23.9·10-s + 11·11-s + 9.35·12-s + 149.·14-s + 3.90·15-s − 5.36·16-s − 83.1·17-s + 120.·18-s + 107.·19-s + 66.7·20-s − 24.3·21-s − 49.9·22-s − 38.9·23-s − 15.5·24-s − 97.1·25-s − 39.5·27-s − 415.·28-s + 74.5·29-s − 17.7·30-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 0.142·3-s + 1.57·4-s + 0.472·5-s − 0.228·6-s − 1.77·7-s − 0.931·8-s − 0.979·9-s − 0.758·10-s + 0.301·11-s + 0.224·12-s + 2.84·14-s + 0.0672·15-s − 0.0838·16-s − 1.18·17-s + 1.57·18-s + 1.29·19-s + 0.745·20-s − 0.252·21-s − 0.484·22-s − 0.353·23-s − 0.132·24-s − 0.777·25-s − 0.281·27-s − 2.80·28-s + 0.477·29-s − 0.107·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.54T + 8T^{2} \) |
| 3 | \( 1 - 0.739T + 27T^{2} \) |
| 5 | \( 1 - 5.27T + 125T^{2} \) |
| 7 | \( 1 + 32.8T + 343T^{2} \) |
| 17 | \( 1 + 83.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 74.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 59.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 406.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 493.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 403.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 545.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 498.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 109.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 573.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 875.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 598.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 840.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 795.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 92.1T + 9.12e5T^{2} \) |
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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
−8.964734061575836824064713804759, −7.78918782983532440433847942489, −7.11101974148183063962507164814, −6.22233334742773619780085169674, −5.80485824384361573441717962475, −4.14156340765163053013812503025, −2.91919539103700682015367414612, −2.34284811423624940922328884663, −0.893227819279806117809988719578, 0,
0.893227819279806117809988719578, 2.34284811423624940922328884663, 2.91919539103700682015367414612, 4.14156340765163053013812503025, 5.80485824384361573441717962475, 6.22233334742773619780085169674, 7.11101974148183063962507164814, 7.78918782983532440433847942489, 8.964734061575836824064713804759
