| L(s) = 1 | − 5.10·2-s + 4.57·3-s + 18.0·4-s − 15.4·5-s − 23.3·6-s − 16.7·7-s − 51.4·8-s − 6.06·9-s + 78.8·10-s + 11·11-s + 82.6·12-s + 85.5·14-s − 70.6·15-s + 117.·16-s − 14.0·17-s + 30.9·18-s + 75.4·19-s − 278.·20-s − 76.6·21-s − 56.1·22-s + 185.·23-s − 235.·24-s + 113.·25-s − 151.·27-s − 302.·28-s − 133.·29-s + 360.·30-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 0.880·3-s + 2.25·4-s − 1.38·5-s − 1.58·6-s − 0.905·7-s − 2.27·8-s − 0.224·9-s + 2.49·10-s + 0.301·11-s + 1.98·12-s + 1.63·14-s − 1.21·15-s + 1.84·16-s − 0.200·17-s + 0.405·18-s + 0.911·19-s − 3.11·20-s − 0.796·21-s − 0.544·22-s + 1.67·23-s − 2.00·24-s + 0.906·25-s − 1.07·27-s − 2.04·28-s − 0.853·29-s + 2.19·30-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 | \( 1 + 5.10T + 8T^{2} \) |
| 3 | \( 1 - 4.57T + 27T^{2} \) |
| 5 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 + 16.7T + 343T^{2} \) |
| 17 | \( 1 + 14.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 166.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 31.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 136.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 96.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 626.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 992.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 142.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 997.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 853.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 715.T + 9.12e5T^{2} \) |
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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
−8.606360701597090692086997323363, −7.85330432519465162948659127178, −7.27518505965651283342049244008, −6.75728567360010899556503255807, −5.45738291631670551053079498531, −3.77275526215413290599360920172, −3.23129762731875699488952787036, −2.28390400457618340152663040039, −0.901859352364359413126035550918, 0,
0.901859352364359413126035550918, 2.28390400457618340152663040039, 3.23129762731875699488952787036, 3.77275526215413290599360920172, 5.45738291631670551053079498531, 6.75728567360010899556503255807, 7.27518505965651283342049244008, 7.85330432519465162948659127178, 8.606360701597090692086997323363
