| L(s) = 1 | + 4.65·2-s + 3·3-s + 13.6·4-s − 5·5-s + 13.9·6-s + 3.38·7-s + 26.4·8-s + 9·9-s − 23.2·10-s + 41.0·12-s − 40.4·13-s + 15.7·14-s − 15·15-s + 13.8·16-s − 136.·17-s + 41.9·18-s − 143.·19-s − 68.4·20-s + 10.1·21-s + 109.·23-s + 79.4·24-s + 25·25-s − 188.·26-s + 27·27-s + 46.3·28-s − 171.·29-s − 69.8·30-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 0.577·3-s + 1.71·4-s − 0.447·5-s + 0.950·6-s + 0.182·7-s + 1.17·8-s + 0.333·9-s − 0.736·10-s + 0.987·12-s − 0.862·13-s + 0.301·14-s − 0.258·15-s + 0.216·16-s − 1.95·17-s + 0.548·18-s − 1.73·19-s − 0.765·20-s + 0.105·21-s + 0.989·23-s + 0.675·24-s + 0.200·25-s − 1.41·26-s + 0.192·27-s + 0.313·28-s − 1.09·29-s − 0.425·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.65T + 8T^{2} \) |
| 7 | \( 1 - 3.38T + 343T^{2} \) |
| 13 | \( 1 + 40.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 221.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 169.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 113.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 98.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 60.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 300.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 716.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 568.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 982.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 31.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 285.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
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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
−8.561404493078211959472035663155, −7.37878524731115187312243474189, −6.81097365669360225520914851204, −6.05234536160865266845838404739, −4.76030705483120700985957420792, −4.53100484357458393239581905553, −3.61965191991204772565776332926, −2.60772482613904090273576025163, −1.99103464750021686277883604502, 0,
1.99103464750021686277883604502, 2.60772482613904090273576025163, 3.61965191991204772565776332926, 4.53100484357458393239581905553, 4.76030705483120700985957420792, 6.05234536160865266845838404739, 6.81097365669360225520914851204, 7.37878524731115187312243474189, 8.561404493078211959472035663155
