| L(s) = 1 | + 0.147·3-s + 5-s − 3.83·7-s − 2.97·9-s + 3.59·11-s + 1.00·13-s + 0.147·15-s − 6.70·17-s + 0.530·19-s − 0.566·21-s + 23-s + 25-s − 0.881·27-s + 10.3·29-s + 6.59·31-s + 0.530·33-s − 3.83·35-s − 0.322·37-s + 0.148·39-s − 2.02·41-s − 11.2·43-s − 2.97·45-s − 8.43·47-s + 7.74·49-s − 0.988·51-s − 5.77·53-s + 3.59·55-s + ⋯ |
| L(s) = 1 | + 0.0851·3-s + 0.447·5-s − 1.45·7-s − 0.992·9-s + 1.08·11-s + 0.279·13-s + 0.0380·15-s − 1.62·17-s + 0.121·19-s − 0.123·21-s + 0.208·23-s + 0.200·25-s − 0.169·27-s + 1.91·29-s + 1.18·31-s + 0.0923·33-s − 0.648·35-s − 0.0529·37-s + 0.0238·39-s − 0.316·41-s − 1.71·43-s − 0.443·45-s − 1.23·47-s + 1.10·49-s − 0.138·51-s − 0.793·53-s + 0.484·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.449058827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449058827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 0.147T + 3T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 - 0.530T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 + 0.322T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 8.43T + 47T^{2} \) |
| 53 | \( 1 + 5.77T + 53T^{2} \) |
| 59 | \( 1 - 5.30T + 59T^{2} \) |
| 61 | \( 1 + 3.30T + 61T^{2} \) |
| 67 | \( 1 + 3.40T + 67T^{2} \) |
| 71 | \( 1 - 1.00T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 8.45T + 89T^{2} \) |
| 97 | \( 1 + 8.61T + 97T^{2} \) |
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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
−8.085856957140826288591124148373, −6.78696559593718025850061205619, −6.46640696594834659844430124798, −6.19745230308691082409134520627, −5.05524854672722134346722775171, −4.34214018650848964020780477057, −3.28906293485400204832376494916, −2.91558386341953010311746608984, −1.87168879695996455669723770334, −0.58473036615085032164069548893,
0.58473036615085032164069548893, 1.87168879695996455669723770334, 2.91558386341953010311746608984, 3.28906293485400204832376494916, 4.34214018650848964020780477057, 5.05524854672722134346722775171, 6.19745230308691082409134520627, 6.46640696594834659844430124798, 6.78696559593718025850061205619, 8.085856957140826288591124148373
