| L(s) = 1 | + 1.78·2-s + 1.19·4-s − 5-s + 7-s − 1.44·8-s − 1.78·10-s + 5.76·11-s + 4.94·13-s + 1.78·14-s − 4.96·16-s + 1.91·17-s + 5.05·19-s − 1.19·20-s + 10.3·22-s + 23-s + 25-s + 8.82·26-s + 1.19·28-s − 5.57·29-s − 9.61·31-s − 5.98·32-s + 3.42·34-s − 35-s + 2.49·37-s + 9.02·38-s + 1.44·40-s − 4.66·41-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.596·4-s − 0.447·5-s + 0.377·7-s − 0.509·8-s − 0.565·10-s + 1.73·11-s + 1.37·13-s + 0.477·14-s − 1.24·16-s + 0.464·17-s + 1.15·19-s − 0.266·20-s + 2.19·22-s + 0.208·23-s + 0.200·25-s + 1.73·26-s + 0.225·28-s − 1.03·29-s − 1.72·31-s − 1.05·32-s + 0.586·34-s − 0.169·35-s + 0.409·37-s + 1.46·38-s + 0.227·40-s − 0.728·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.437171611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.437171611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 + 4.66T + 41T^{2} \) |
| 43 | \( 1 - 6.84T + 43T^{2} \) |
| 47 | \( 1 + 2.41T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 - 9.83T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 + 4.46T + 79T^{2} \) |
| 83 | \( 1 - 2.52T + 83T^{2} \) |
| 89 | \( 1 + 0.000952T + 89T^{2} \) |
| 97 | \( 1 + 4.66T + 97T^{2} \) |
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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
−7.77076721779142655505779622912, −7.00059508585860360246833075133, −6.38068563734314200731774332286, −5.64286823771242834583592329876, −5.14167587480679886373077631058, −4.07824281039194485326714150898, −3.75386634896028644368818597321, −3.23329032771877543121835792656, −1.86847155803201193568173795472, −0.939222666250022427750556303901,
0.939222666250022427750556303901, 1.86847155803201193568173795472, 3.23329032771877543121835792656, 3.75386634896028644368818597321, 4.07824281039194485326714150898, 5.14167587480679886373077631058, 5.64286823771242834583592329876, 6.38068563734314200731774332286, 7.00059508585860360246833075133, 7.77076721779142655505779622912
