| L(s) = 1 | + 3.23·3-s − 2·5-s + 3.23·7-s + 7.47·9-s − 3.23·11-s + 4.47·13-s − 6.47·15-s + 17-s − 2.47·19-s + 10.4·21-s + 3.23·23-s − 25-s + 14.4·27-s − 2·29-s − 3.23·31-s − 10.4·33-s − 6.47·35-s − 6.94·37-s + 14.4·39-s + 2·41-s + 10.4·43-s − 14.9·45-s − 4.94·47-s + 3.47·49-s + 3.23·51-s + 2·53-s + 6.47·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 0.894·5-s + 1.22·7-s + 2.49·9-s − 0.975·11-s + 1.24·13-s − 1.67·15-s + 0.242·17-s − 0.567·19-s + 2.28·21-s + 0.674·23-s − 0.200·25-s + 2.78·27-s − 0.371·29-s − 0.581·31-s − 1.82·33-s − 1.09·35-s − 1.14·37-s + 2.31·39-s + 0.312·41-s + 1.59·43-s − 2.22·45-s − 0.721·47-s + 0.496·49-s + 0.453·51-s + 0.274·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.063738882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.063738882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 4.76T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.646355245539427118435441733275, −8.664452091575302382248579030212, −8.276950079405252420792680579380, −7.73926603612761120578146469520, −6.99690934082156437392070881139, −5.36681748032918385279753370648, −4.24299421281791484386856129471, −3.64634861849280116941857159367, −2.57177951014745590833816631740, −1.48970995245309538523517247227,
1.48970995245309538523517247227, 2.57177951014745590833816631740, 3.64634861849280116941857159367, 4.24299421281791484386856129471, 5.36681748032918385279753370648, 6.99690934082156437392070881139, 7.73926603612761120578146469520, 8.276950079405252420792680579380, 8.664452091575302382248579030212, 9.646355245539427118435441733275
