| L(s) = 1 | + 3.76·2-s − 4.70·3-s + 6.16·4-s − 20.6·5-s − 17.7·6-s + 28.6·7-s − 6.91·8-s − 4.84·9-s − 77.5·10-s + 11·11-s − 29.0·12-s + 107.·14-s + 96.9·15-s − 75.3·16-s + 50.0·17-s − 18.2·18-s − 5.38·19-s − 126.·20-s − 134.·21-s + 41.3·22-s + 183.·23-s + 32.5·24-s + 299.·25-s + 149.·27-s + 176.·28-s − 200.·29-s + 364.·30-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.905·3-s + 0.770·4-s − 1.84·5-s − 1.20·6-s + 1.54·7-s − 0.305·8-s − 0.179·9-s − 2.45·10-s + 0.301·11-s − 0.697·12-s + 2.05·14-s + 1.66·15-s − 1.17·16-s + 0.714·17-s − 0.238·18-s − 0.0649·19-s − 1.41·20-s − 1.40·21-s + 0.401·22-s + 1.66·23-s + 0.276·24-s + 2.39·25-s + 1.06·27-s + 1.19·28-s − 1.28·29-s + 2.22·30-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 | \( 1 - 3.76T + 8T^{2} \) |
| 3 | \( 1 + 4.70T + 27T^{2} \) |
| 5 | \( 1 + 20.6T + 125T^{2} \) |
| 7 | \( 1 - 28.6T + 343T^{2} \) |
| 17 | \( 1 - 50.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.38T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 19.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 359.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 261.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 725.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 381.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 68.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 142.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 394.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 519.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 248.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 848.T + 9.12e5T^{2} \) |
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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
−8.428138630342471109796563258180, −7.39958860483808541711898892952, −6.93039464954615370897996449640, −5.57762587647717936461968572066, −5.18622875510026158952828167877, −4.42274637533817708376909103387, −3.80345811730959739130125300657, −2.85925412198694620138403980953, −1.15830332928477601461727025921, 0,
1.15830332928477601461727025921, 2.85925412198694620138403980953, 3.80345811730959739130125300657, 4.42274637533817708376909103387, 5.18622875510026158952828167877, 5.57762587647717936461968572066, 6.93039464954615370897996449640, 7.39958860483808541711898892952, 8.428138630342471109796563258180
