| L(s) = 1 | + 1.12·2-s + 9.49·3-s − 6.72·4-s − 7.22·5-s + 10.7·6-s + 1.17·7-s − 16.6·8-s + 63.1·9-s − 8.14·10-s + 11·11-s − 63.9·12-s + 1.32·14-s − 68.5·15-s + 35.1·16-s − 101.·17-s + 71.2·18-s + 23.3·19-s + 48.5·20-s + 11.1·21-s + 12.4·22-s − 29.6·23-s − 157.·24-s − 72.8·25-s + 343.·27-s − 7.91·28-s − 214.·29-s − 77.3·30-s + ⋯ |
| L(s) = 1 | + 0.398·2-s + 1.82·3-s − 0.841·4-s − 0.645·5-s + 0.728·6-s + 0.0634·7-s − 0.733·8-s + 2.34·9-s − 0.257·10-s + 0.301·11-s − 1.53·12-s + 0.0253·14-s − 1.18·15-s + 0.548·16-s − 1.45·17-s + 0.933·18-s + 0.281·19-s + 0.543·20-s + 0.116·21-s + 0.120·22-s − 0.268·23-s − 1.34·24-s − 0.582·25-s + 2.45·27-s − 0.0533·28-s − 1.37·29-s − 0.470·30-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 | \( 1 - 1.12T + 8T^{2} \) |
| 3 | \( 1 - 9.49T + 27T^{2} \) |
| 5 | \( 1 + 7.22T + 125T^{2} \) |
| 7 | \( 1 - 1.17T + 343T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 29.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 144.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 77.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 345.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 764.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 782.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 336.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 691.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 7.42T + 4.93e5T^{2} \) |
| 83 | \( 1 - 708.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−8.437557920249677439788418704860, −7.979261145810153923934641711247, −7.18357656811340553459276724035, −6.13934682796044678702557029970, −4.72822981233896241727033472672, −4.18431285655477886035186044988, −3.51963906992123287008757766116, −2.69169908151090858083310861743, −1.56750932477242117220030514399, 0,
1.56750932477242117220030514399, 2.69169908151090858083310861743, 3.51963906992123287008757766116, 4.18431285655477886035186044988, 4.72822981233896241727033472672, 6.13934682796044678702557029970, 7.18357656811340553459276724035, 7.979261145810153923934641711247, 8.437557920249677439788418704860
