| L(s) = 1 | − 2.49·2-s + 1.54·3-s + 4.23·4-s − 3.85·6-s + 0.953·7-s − 5.58·8-s − 0.618·9-s + 2·11-s + 6.53·12-s + 4.99·13-s − 2.38·14-s + 5.47·16-s + 3.08·17-s + 1.54·18-s + 2.76·19-s + 1.47·21-s − 4.99·22-s − 4.04·23-s − 8.61·24-s − 12.4·26-s − 5.58·27-s + 4.04·28-s − 5.85·29-s + 2·31-s − 2.49·32-s + 3.08·33-s − 7.70·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.891·3-s + 2.11·4-s − 1.57·6-s + 0.360·7-s − 1.97·8-s − 0.206·9-s + 0.603·11-s + 1.88·12-s + 1.38·13-s − 0.636·14-s + 1.36·16-s + 0.748·17-s + 0.363·18-s + 0.634·19-s + 0.321·21-s − 1.06·22-s − 0.842·23-s − 1.75·24-s − 2.44·26-s − 1.07·27-s + 0.763·28-s − 1.08·29-s + 0.359·31-s − 0.441·32-s + 0.537·33-s − 1.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6902470835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6902470835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 - 1.54T + 3T^{2} \) |
| 7 | \( 1 - 0.953T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 - 3.08T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 9.62T + 43T^{2} \) |
| 47 | \( 1 - 6.53T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8.80T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−13.60781578717060663689731556098, −11.88706353935410087322819753227, −11.07866769582557529372233522673, −9.930797908371468858614118209997, −8.991938145131980755234269314889, −8.325216480590417468525514395805, −7.45967420789771327974830432910, −6.02989432854471149478376449495, −3.41358495683600403314810394032, −1.64895008755488486949195403478,
1.64895008755488486949195403478, 3.41358495683600403314810394032, 6.02989432854471149478376449495, 7.45967420789771327974830432910, 8.325216480590417468525514395805, 8.991938145131980755234269314889, 9.930797908371468858614118209997, 11.07866769582557529372233522673, 11.88706353935410087322819753227, 13.60781578717060663689731556098
