| L(s) = 1 | + 2-s + 2.23·3-s + 4-s + 5-s + 2.23·6-s + 1.43·7-s + 8-s + 2.00·9-s + 10-s + 0.315·11-s + 2.23·12-s + 1.43·14-s + 2.23·15-s + 16-s − 1.69·17-s + 2.00·18-s + 5.83·19-s + 20-s + 3.20·21-s + 0.315·22-s − 9.32·23-s + 2.23·24-s + 25-s − 2.22·27-s + 1.43·28-s + 7.09·29-s + 2.23·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.447·5-s + 0.913·6-s + 0.541·7-s + 0.353·8-s + 0.668·9-s + 0.316·10-s + 0.0952·11-s + 0.645·12-s + 0.382·14-s + 0.577·15-s + 0.250·16-s − 0.412·17-s + 0.472·18-s + 1.33·19-s + 0.223·20-s + 0.699·21-s + 0.0673·22-s − 1.94·23-s + 0.456·24-s + 0.200·25-s − 0.428·27-s + 0.270·28-s + 1.31·29-s + 0.408·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.699464452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.699464452\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 - 0.315T + 11T^{2} \) |
| 17 | \( 1 + 1.69T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 23 | \( 1 + 9.32T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 1.48T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 - 0.971T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 - 8.83T + 73T^{2} \) |
| 79 | \( 1 + 6.80T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−9.310468318302112658863737197982, −8.399997676157438479567152903418, −7.909067424080073954303005310281, −6.99509693743342865015461071128, −6.05461984475999528184383191792, −5.14252606370394161741627106541, −4.21880150109581798059492709070, −3.31704446082763327899016019653, −2.46964067366439353107585936420, −1.57786949822264456736851835422,
1.57786949822264456736851835422, 2.46964067366439353107585936420, 3.31704446082763327899016019653, 4.21880150109581798059492709070, 5.14252606370394161741627106541, 6.05461984475999528184383191792, 6.99509693743342865015461071128, 7.909067424080073954303005310281, 8.399997676157438479567152903418, 9.310468318302112658863737197982
