| L(s) = 1 | + 3.94·5-s + 7-s − 0.942·11-s − 13-s − 5.60·17-s + 1.28·19-s + 4.66·23-s + 10.5·25-s + 7.66·29-s + 7.82·31-s + 3.94·35-s − 9.82·37-s − 0.942·41-s + 9.26·43-s + 5.28·47-s + 49-s + 9.22·53-s − 3.71·55-s + 9.54·59-s − 10.5·61-s − 3.94·65-s − 1.71·67-s − 3.54·71-s + 2.71·73-s − 0.942·77-s − 16.3·79-s + 0.396·83-s + ⋯ |
| L(s) = 1 | + 1.76·5-s + 0.377·7-s − 0.284·11-s − 0.277·13-s − 1.35·17-s + 0.294·19-s + 0.971·23-s + 2.10·25-s + 1.42·29-s + 1.40·31-s + 0.666·35-s − 1.61·37-s − 0.147·41-s + 1.41·43-s + 0.770·47-s + 0.142·49-s + 1.26·53-s − 0.501·55-s + 1.24·59-s − 1.35·61-s − 0.489·65-s − 0.209·67-s − 0.420·71-s + 0.318·73-s − 0.107·77-s − 1.84·79-s + 0.0435·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.625669282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.625669282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.94T + 5T^{2} \) |
| 11 | \( 1 + 0.942T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 - 4.66T + 23T^{2} \) |
| 29 | \( 1 - 7.66T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 + 0.942T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 2.71T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 0.396T + 83T^{2} \) |
| 89 | \( 1 + 5.50T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.864300791330968103539285390010, −8.649539729398477138502746442926, −7.27750147617083737871357181065, −6.64384023419566450006948882387, −5.86810674829300134222377277796, −5.10465701277197981462820352526, −4.43929209960523638907577042458, −2.84344383354613276560775416923, −2.25134314512312011179343778810, −1.12904004686006969109738342785,
1.12904004686006969109738342785, 2.25134314512312011179343778810, 2.84344383354613276560775416923, 4.43929209960523638907577042458, 5.10465701277197981462820352526, 5.86810674829300134222377277796, 6.64384023419566450006948882387, 7.27750147617083737871357181065, 8.649539729398477138502746442926, 8.864300791330968103539285390010
