| L(s) = 1 | − 2.83·2-s + 8.17·3-s + 0.0460·4-s − 2.30·5-s − 23.1·6-s + 3.15·7-s + 22.5·8-s + 39.8·9-s + 6.55·10-s − 11·11-s + 0.376·12-s − 8.95·14-s − 18.8·15-s − 64.3·16-s − 127.·17-s − 113.·18-s + 108.·19-s − 0.106·20-s + 25.8·21-s + 31.2·22-s − 19.0·23-s + 184.·24-s − 119.·25-s + 105.·27-s + 0.145·28-s + 197.·29-s + 53.5·30-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.57·3-s + 0.00576·4-s − 0.206·5-s − 1.57·6-s + 0.170·7-s + 0.997·8-s + 1.47·9-s + 0.207·10-s − 0.301·11-s + 0.00906·12-s − 0.171·14-s − 0.325·15-s − 1.00·16-s − 1.82·17-s − 1.48·18-s + 1.30·19-s − 0.00118·20-s + 0.268·21-s + 0.302·22-s − 0.172·23-s + 1.56·24-s − 0.957·25-s + 0.749·27-s + 0.000982·28-s + 1.26·29-s + 0.325·30-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 | \( 1 + 2.83T + 8T^{2} \) |
| 3 | \( 1 - 8.17T + 27T^{2} \) |
| 5 | \( 1 + 2.30T + 125T^{2} \) |
| 7 | \( 1 - 3.15T + 343T^{2} \) |
| 17 | \( 1 + 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 197.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 326.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 406.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 625.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 583.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 578.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 103.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 660.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−8.647560231611616672387677233888, −7.946420339571388207867923199666, −7.40045109058576079872149631763, −6.50006366213834006974200185846, −4.92598468981180686210154266998, −4.24873587661639734397123478588, −3.20444003096770659410597714698, −2.27986668673898367855715885948, −1.36381245306145200242025705524, 0,
1.36381245306145200242025705524, 2.27986668673898367855715885948, 3.20444003096770659410597714698, 4.24873587661639734397123478588, 4.92598468981180686210154266998, 6.50006366213834006974200185846, 7.40045109058576079872149631763, 7.946420339571388207867923199666, 8.647560231611616672387677233888
