| L(s) = 1 | − 2-s + 3·3-s − 4-s − 3·6-s − 3·7-s + 3·8-s + 6·9-s − 3·12-s + 4·13-s + 3·14-s − 16-s − 6·18-s − 4·19-s − 9·21-s + 8·23-s + 9·24-s − 4·26-s + 9·27-s + 3·28-s − 6·29-s − 2·31-s − 5·32-s − 6·36-s + 8·37-s + 4·38-s + 12·39-s + 5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.22·6-s − 1.13·7-s + 1.06·8-s + 2·9-s − 0.866·12-s + 1.10·13-s + 0.801·14-s − 1/4·16-s − 1.41·18-s − 0.917·19-s − 1.96·21-s + 1.66·23-s + 1.83·24-s − 0.784·26-s + 1.73·27-s + 0.566·28-s − 1.11·29-s − 0.359·31-s − 0.883·32-s − 36-s + 1.31·37-s + 0.648·38-s + 1.92·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.957764678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.957764678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.787060254601332760003741688288, −8.263976819145425469141106348915, −7.47660059173941161195224407791, −6.83431260685648222910772054075, −5.79830221528795947612569327096, −4.45971739155118948804825345937, −3.78259424196032201632404802726, −3.11004172006035929044721752931, −2.09993952223754336425071201934, −0.903515928290081006220007479275,
0.903515928290081006220007479275, 2.09993952223754336425071201934, 3.11004172006035929044721752931, 3.78259424196032201632404802726, 4.45971739155118948804825345937, 5.79830221528795947612569327096, 6.83431260685648222910772054075, 7.47660059173941161195224407791, 8.263976819145425469141106348915, 8.787060254601332760003741688288
