| L(s) = 1 | + 2.60·2-s + 4.77·4-s − 0.851·5-s + 3.04·7-s + 7.23·8-s − 2.21·10-s + 11-s + 1.25·13-s + 7.92·14-s + 9.28·16-s + 7.73·17-s + 6.82·19-s − 4.06·20-s + 2.60·22-s − 8.40·23-s − 4.27·25-s + 3.27·26-s + 14.5·28-s − 1.09·29-s − 2.03·31-s + 9.70·32-s + 20.1·34-s − 2.59·35-s − 6.32·37-s + 17.7·38-s − 6.16·40-s + 9.82·41-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.38·4-s − 0.380·5-s + 1.15·7-s + 2.55·8-s − 0.701·10-s + 0.301·11-s + 0.348·13-s + 2.11·14-s + 2.32·16-s + 1.87·17-s + 1.56·19-s − 0.909·20-s + 0.555·22-s − 1.75·23-s − 0.855·25-s + 0.641·26-s + 2.74·28-s − 0.203·29-s − 0.365·31-s + 1.71·32-s + 3.45·34-s − 0.437·35-s − 1.03·37-s + 2.88·38-s − 0.974·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.060391171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.060391171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 5 | \( 1 + 0.851T + 5T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 13 | \( 1 - 1.25T + 13T^{2} \) |
| 17 | \( 1 - 7.73T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 - 9.82T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 + 1.11T + 47T^{2} \) |
| 53 | \( 1 - 4.72T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 - 1.27T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 3.62T + 89T^{2} \) |
| 97 | \( 1 + 0.113T + 97T^{2} \) |
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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
−7.59392821619711965180114219748, −7.52416058947393354601429564685, −6.33710198335406585700577670598, −5.59993448180496592103697304302, −5.29264966358271195603970641939, −4.40286381609584622625240477341, −3.69686068778247814688927988663, −3.24435530721432332159736032175, −2.03123276169935280209489302001, −1.29745395990437154488459362457,
1.29745395990437154488459362457, 2.03123276169935280209489302001, 3.24435530721432332159736032175, 3.69686068778247814688927988663, 4.40286381609584622625240477341, 5.29264966358271195603970641939, 5.59993448180496592103697304302, 6.33710198335406585700577670598, 7.52416058947393354601429564685, 7.59392821619711965180114219748
