| L(s) = 1 | − 1.64·5-s + 7-s + 2·11-s − 4·13-s − 5.64·17-s + 19-s − 6·23-s − 2.29·25-s + 0.354·29-s − 9.29·31-s − 1.64·35-s − 5.29·37-s + 12.5·41-s − 2·43-s + 7.64·47-s + 49-s + 6.93·53-s − 3.29·55-s − 6.58·59-s + 13.2·61-s + 6.58·65-s − 6.58·67-s + 5.64·71-s + 1.29·73-s + 2·77-s + 0.708·79-s + 1.06·83-s + ⋯ |
| L(s) = 1 | − 0.736·5-s + 0.377·7-s + 0.603·11-s − 1.10·13-s − 1.36·17-s + 0.229·19-s − 1.25·23-s − 0.458·25-s + 0.0657·29-s − 1.66·31-s − 0.278·35-s − 0.869·37-s + 1.96·41-s − 0.304·43-s + 1.11·47-s + 0.142·49-s + 0.952·53-s − 0.443·55-s − 0.857·59-s + 1.70·61-s + 0.816·65-s − 0.804·67-s + 0.670·71-s + 0.151·73-s + 0.227·77-s + 0.0797·79-s + 0.116·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099832719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099832719\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.64T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 5.64T + 17T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 0.354T + 29T^{2} \) |
| 31 | \( 1 + 9.29T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 6.93T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 - 5.64T + 71T^{2} \) |
| 73 | \( 1 - 1.29T + 73T^{2} \) |
| 79 | \( 1 - 0.708T + 79T^{2} \) |
| 83 | \( 1 - 1.06T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4.58T + 97T^{2} \) |
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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
−7.42123977774045377443152477608, −7.34251128776754266461011117422, −6.35763427696435639169456475604, −5.62987357478536791566424374195, −4.82307016569430279930741481353, −4.10904536613812480863525333843, −3.69515925652149566266726443930, −2.44520358434183380005776173201, −1.88123004508026138895897625020, −0.48068620101203434500420582618,
0.48068620101203434500420582618, 1.88123004508026138895897625020, 2.44520358434183380005776173201, 3.69515925652149566266726443930, 4.10904536613812480863525333843, 4.82307016569430279930741481353, 5.62987357478536791566424374195, 6.35763427696435639169456475604, 7.34251128776754266461011117422, 7.42123977774045377443152477608
