| L(s) = 1 | − 1.93·3-s + 2.93·5-s − 4.68·7-s + 0.745·9-s + 0.762·11-s − 1.76·13-s − 5.68·15-s + 3.36·17-s + 7.36·19-s + 9.06·21-s − 3.25·23-s + 3.61·25-s + 4.36·27-s + 3.25·29-s − 3.06·31-s − 1.47·33-s − 13.7·35-s + 37-s + 3.41·39-s − 7.42·41-s − 12.2·43-s + 2.18·45-s − 0.302·47-s + 14.9·49-s − 6.50·51-s − 5.53·53-s + 2.23·55-s + ⋯ |
| L(s) = 1 | − 1.11·3-s + 1.31·5-s − 1.76·7-s + 0.248·9-s + 0.229·11-s − 0.488·13-s − 1.46·15-s + 0.815·17-s + 1.68·19-s + 1.97·21-s − 0.678·23-s + 0.723·25-s + 0.839·27-s + 0.604·29-s − 0.550·31-s − 0.256·33-s − 2.32·35-s + 0.164·37-s + 0.546·39-s − 1.15·41-s − 1.86·43-s + 0.326·45-s − 0.0440·47-s + 2.13·49-s − 0.911·51-s − 0.760·53-s + 0.301·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 - 0.762T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + 3.25T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 41 | \( 1 + 7.42T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 0.302T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 0.173T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 4.61T + 79T^{2} \) |
| 83 | \( 1 - 3.53T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.902386820949328232085049668707, −7.58386915294965718612748495956, −6.65695905187835746597161811998, −6.24303256703266197267961835147, −5.55734847331483286213249208269, −5.02027819319793879495335637939, −3.50115944702038849552515004838, −2.79363622965920750775064069811, −1.37093618209581543756882175601, 0,
1.37093618209581543756882175601, 2.79363622965920750775064069811, 3.50115944702038849552515004838, 5.02027819319793879495335637939, 5.55734847331483286213249208269, 6.24303256703266197267961835147, 6.65695905187835746597161811998, 7.58386915294965718612748495956, 8.902386820949328232085049668707
