| L(s) = 1 | − 1.49·2-s − 8.91·3-s − 5.77·4-s − 12.8·5-s + 13.2·6-s + 33.8·7-s + 20.5·8-s + 52.4·9-s + 19.2·10-s + 11·11-s + 51.5·12-s − 50.5·14-s + 114.·15-s + 15.6·16-s − 129.·17-s − 78.1·18-s − 74.1·19-s + 74.5·20-s − 302.·21-s − 16.3·22-s + 167.·23-s − 183.·24-s + 41.1·25-s − 227.·27-s − 195.·28-s − 95.2·29-s − 171.·30-s + ⋯ |
| L(s) = 1 | − 0.526·2-s − 1.71·3-s − 0.722·4-s − 1.15·5-s + 0.903·6-s + 1.83·7-s + 0.907·8-s + 1.94·9-s + 0.607·10-s + 0.301·11-s + 1.23·12-s − 0.964·14-s + 1.97·15-s + 0.244·16-s − 1.84·17-s − 1.02·18-s − 0.894·19-s + 0.832·20-s − 3.13·21-s − 0.158·22-s + 1.51·23-s − 1.55·24-s + 0.329·25-s − 1.61·27-s − 1.32·28-s − 0.610·29-s − 1.04·30-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 | \( 1 + 1.49T + 8T^{2} \) |
| 3 | \( 1 + 8.91T + 27T^{2} \) |
| 5 | \( 1 + 12.8T + 125T^{2} \) |
| 7 | \( 1 - 33.8T + 343T^{2} \) |
| 17 | \( 1 + 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 34.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 332.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 229.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 541.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 59.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 670.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 27.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 173.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 183.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−8.641682566971627344111622617300, −7.55144476653713955755190433045, −7.17181424621977622469231275677, −5.99868894114720544076167466402, −4.95322775558660427976883974888, −4.56645641686443965463072202533, −4.05222173364661359925618343403, −1.82859836247691763705849552800, −0.842480519357318530248792789015, 0,
0.842480519357318530248792789015, 1.82859836247691763705849552800, 4.05222173364661359925618343403, 4.56645641686443965463072202533, 4.95322775558660427976883974888, 5.99868894114720544076167466402, 7.17181424621977622469231275677, 7.55144476653713955755190433045, 8.641682566971627344111622617300
