| L(s) = 1 | + 0.0913·2-s + 1.80·3-s − 1.99·4-s − 5-s + 0.165·6-s + 7-s − 0.364·8-s + 0.275·9-s − 0.0913·10-s − 0.144·11-s − 3.60·12-s − 4.94·13-s + 0.0913·14-s − 1.80·15-s + 3.95·16-s + 5.07·17-s + 0.0251·18-s − 4.55·19-s + 1.99·20-s + 1.80·21-s − 0.0132·22-s + 3.63·23-s − 0.659·24-s + 25-s − 0.451·26-s − 4.93·27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | + 0.0645·2-s + 1.04·3-s − 0.995·4-s − 0.447·5-s + 0.0674·6-s + 0.377·7-s − 0.128·8-s + 0.0919·9-s − 0.0288·10-s − 0.0437·11-s − 1.04·12-s − 1.37·13-s + 0.0244·14-s − 0.467·15-s + 0.987·16-s + 1.23·17-s + 0.00593·18-s − 1.04·19-s + 0.445·20-s + 0.394·21-s − 0.00282·22-s + 0.757·23-s − 0.134·24-s + 0.200·25-s − 0.0886·26-s − 0.948·27-s − 0.376·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0913T + 2T^{2} \) |
| 3 | \( 1 - 1.80T + 3T^{2} \) |
| 11 | \( 1 + 0.144T + 11T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.80T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 0.732T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 - 0.733T + 97T^{2} \) |
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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
−7.70232484032739085140225899483, −7.12711452492050070060230030808, −5.95335453806231855581277431341, −5.23268074910143139225838540943, −4.50767884567570596376039726803, −3.97648613228679870672074967133, −3.02594595956054731903594628768, −2.53857092317292048533214073129, −1.23728580705629956885006984965, 0,
1.23728580705629956885006984965, 2.53857092317292048533214073129, 3.02594595956054731903594628768, 3.97648613228679870672074967133, 4.50767884567570596376039726803, 5.23268074910143139225838540943, 5.95335453806231855581277431341, 7.12711452492050070060230030808, 7.70232484032739085140225899483
