| L(s) = 1 | − 2-s + 3.23·3-s + 4-s − 3.23·6-s − 1.61·7-s − 8-s + 7.47·9-s + 3.38·11-s + 3.23·12-s − 2.61·13-s + 1.61·14-s + 16-s − 2.47·17-s − 7.47·18-s − 3.61·19-s − 5.23·21-s − 3.38·22-s + 0.145·23-s − 3.23·24-s + 2.61·26-s + 14.4·27-s − 1.61·28-s + 2.76·29-s − 5.23·31-s − 32-s + 10.9·33-s + 2.47·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.86·3-s + 0.5·4-s − 1.32·6-s − 0.611·7-s − 0.353·8-s + 2.49·9-s + 1.01·11-s + 0.934·12-s − 0.726·13-s + 0.432·14-s + 0.250·16-s − 0.599·17-s − 1.76·18-s − 0.830·19-s − 1.14·21-s − 0.721·22-s + 0.0304·23-s − 0.660·24-s + 0.513·26-s + 2.78·27-s − 0.305·28-s + 0.513·29-s − 0.940·31-s − 0.176·32-s + 1.90·33-s + 0.423·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567205781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567205781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 - 0.145T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.23T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 8.18T + 97T^{2} \) |
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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
−12.25373800315389901110499956124, −10.74918565806707784410707027856, −9.668475873277787247465169119999, −9.171708519184234308202210348606, −8.386705907813806719528446610267, −7.35206506573418131255305753615, −6.53098679144424566222188949402, −4.27538892383386206222660915074, −3.11385368846793388235285496868, −1.92616098242808541381348615011,
1.92616098242808541381348615011, 3.11385368846793388235285496868, 4.27538892383386206222660915074, 6.53098679144424566222188949402, 7.35206506573418131255305753615, 8.386705907813806719528446610267, 9.171708519184234308202210348606, 9.668475873277787247465169119999, 10.74918565806707784410707027856, 12.25373800315389901110499956124
