| L(s) = 1 | + 5-s − 1.23·7-s − 2·11-s + 4.47·13-s + 4.47·17-s − 4.47·19-s − 9.23·23-s + 25-s + 2·29-s + 2.47·31-s − 1.23·35-s − 10.9·37-s − 3.52·41-s − 5.70·43-s − 2.76·47-s − 5.47·49-s + 8.47·53-s − 2·55-s + 0.472·59-s − 6·61-s + 4.47·65-s + 5.70·67-s + 6.47·71-s − 4.47·73-s + 2.47·77-s + 4.94·79-s + 9.70·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.467·7-s − 0.603·11-s + 1.24·13-s + 1.08·17-s − 1.02·19-s − 1.92·23-s + 0.200·25-s + 0.371·29-s + 0.444·31-s − 0.208·35-s − 1.79·37-s − 0.550·41-s − 0.870·43-s − 0.403·47-s − 0.781·49-s + 1.16·53-s − 0.269·55-s + 0.0614·59-s − 0.768·61-s + 0.554·65-s + 0.697·67-s + 0.768·71-s − 0.523·73-s + 0.281·77-s + 0.556·79-s + 1.06·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 9.23T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 0.472T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 + 7.52T + 97T^{2} \) |
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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
−8.063510440792229170178722587389, −6.86325153341531721408987798041, −6.31884392410284253779848799898, −5.70526904913018600231547578050, −4.97628158426881351580790421428, −3.89234098861926161640897781379, −3.35187195987591370128456893009, −2.28520827124949966873677443601, −1.41969169944177301690637188479, 0,
1.41969169944177301690637188479, 2.28520827124949966873677443601, 3.35187195987591370128456893009, 3.89234098861926161640897781379, 4.97628158426881351580790421428, 5.70526904913018600231547578050, 6.31884392410284253779848799898, 6.86325153341531721408987798041, 8.063510440792229170178722587389
