L(s) = 1 | + 1.61·2-s + 0.618·4-s − 5-s − 7-s − 2.23·8-s − 1.61·10-s + 5.47·11-s + 5.09·13-s − 1.61·14-s − 4.85·16-s + 4.38·17-s + 2.61·19-s − 0.618·20-s + 8.85·22-s + 6.61·23-s + 25-s + 8.23·26-s − 0.618·28-s − 3.85·29-s + 3·31-s − 3.38·32-s + 7.09·34-s + 35-s − 3·37-s + 4.23·38-s + 2.23·40-s + 1.61·41-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.309·4-s − 0.447·5-s − 0.377·7-s − 0.790·8-s − 0.511·10-s + 1.64·11-s + 1.41·13-s − 0.432·14-s − 1.21·16-s + 1.06·17-s + 0.600·19-s − 0.138·20-s + 1.88·22-s + 1.37·23-s + 0.200·25-s + 1.61·26-s − 0.116·28-s − 0.715·29-s + 0.538·31-s − 0.597·32-s + 1.21·34-s + 0.169·35-s − 0.493·37-s + 0.687·38-s + 0.353·40-s + 0.252·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.640443000\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.640443000\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 + 3.85T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 1.61T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 - 4.38T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 2.14T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 4.52T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−10.03633210691854259383451190882, −9.083105148706136042416193519449, −8.562730392232975027846106217379, −7.20486014744947811110534132345, −6.41657940988785949531951197400, −5.67458381309765849341976589501, −4.63245396726622019814592377168, −3.59487385431958100572888589503, −3.29458834584304556554449057791, −1.21663621455259393726745348875,
1.21663621455259393726745348875, 3.29458834584304556554449057791, 3.59487385431958100572888589503, 4.63245396726622019814592377168, 5.67458381309765849341976589501, 6.41657940988785949531951197400, 7.20486014744947811110534132345, 8.562730392232975027846106217379, 9.083105148706136042416193519449, 10.03633210691854259383451190882