| L(s) = 1 | − 5-s + 4.83·7-s + 2.25·11-s − 13-s − 6.32·17-s + 2.58·19-s + 4.83·23-s + 25-s − 9.09·29-s + 0.510·31-s − 4.83·35-s + 4.25·37-s − 0.255·41-s − 9.16·43-s − 4.51·47-s + 16.4·49-s + 9.93·53-s − 2.25·55-s + 14.1·59-s + 9.93·61-s + 65-s + 13.1·67-s − 5.74·71-s + 2.90·73-s + 10.9·77-s − 5.74·79-s + 10.1·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.82·7-s + 0.679·11-s − 0.277·13-s − 1.53·17-s + 0.592·19-s + 1.00·23-s + 0.200·25-s − 1.68·29-s + 0.0916·31-s − 0.818·35-s + 0.699·37-s − 0.0398·41-s − 1.39·43-s − 0.657·47-s + 2.34·49-s + 1.36·53-s − 0.304·55-s + 1.84·59-s + 1.27·61-s + 0.124·65-s + 1.60·67-s − 0.681·71-s + 0.340·73-s + 1.24·77-s − 0.646·79-s + 1.11·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.466361395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466361395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 + 9.09T + 29T^{2} \) |
| 31 | \( 1 - 0.510T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 + 0.255T + 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 - 9.93T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 9.93T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 5.74T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 + 5.74T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.62825964928972092166575074733, −7.16496860870472444853294133950, −6.49963725903262689480146310455, −5.35134222682283853940187382794, −5.01280920999931799042632129503, −4.22606261695811123631444909171, −3.65969363504919434099117109245, −2.41188420711037434972321787037, −1.76095888701651119804812048287, −0.77971306716276788904907198298,
0.77971306716276788904907198298, 1.76095888701651119804812048287, 2.41188420711037434972321787037, 3.65969363504919434099117109245, 4.22606261695811123631444909171, 5.01280920999931799042632129503, 5.35134222682283853940187382794, 6.49963725903262689480146310455, 7.16496860870472444853294133950, 7.62825964928972092166575074733
