| L(s) = 1 | − 5-s − 4.73·7-s − 4.19·11-s − 1.26·13-s + 6.92·17-s + 19-s + 6·23-s + 25-s + 10.1·29-s − 2·31-s + 4.73·35-s + 2.73·37-s − 11.6·41-s − 4.73·43-s + 10·47-s + 15.3·49-s + 4.19·55-s + 9.46·59-s + 5.46·61-s + 1.26·65-s − 14.9·67-s − 12.3·71-s + 0.928·73-s + 19.8·77-s − 8·79-s − 8.53·83-s − 6.92·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.78·7-s − 1.26·11-s − 0.351·13-s + 1.68·17-s + 0.229·19-s + 1.25·23-s + 0.200·25-s + 1.89·29-s − 0.359·31-s + 0.799·35-s + 0.449·37-s − 1.82·41-s − 0.721·43-s + 1.45·47-s + 2.19·49-s + 0.565·55-s + 1.23·59-s + 0.699·61-s + 0.157·65-s − 1.82·67-s − 1.47·71-s + 0.108·73-s + 2.26·77-s − 0.900·79-s − 0.936·83-s − 0.751·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−7.41878028816641078108945960248, −7.08020834164659534722953366510, −6.23516895237627120478645743339, −5.48382255323977625133337996697, −4.88184237790997095228181877713, −3.77803383958968206510459517733, −2.99565040972473784492079679255, −2.76223583260354349877241079365, −1.03699444365904348453961048063, 0,
1.03699444365904348453961048063, 2.76223583260354349877241079365, 2.99565040972473784492079679255, 3.77803383958968206510459517733, 4.88184237790997095228181877713, 5.48382255323977625133337996697, 6.23516895237627120478645743339, 7.08020834164659534722953366510, 7.41878028816641078108945960248
