| L(s) = 1 | − 1.86·2-s + 1.48·4-s − 1.48·5-s − 4.98·7-s + 0.955·8-s + 2.76·10-s − 1.75·11-s − 5.37·13-s + 9.30·14-s − 4.76·16-s + 3.34·17-s + 2.16·19-s − 2.20·20-s + 3.28·22-s + 7.28·23-s − 2.80·25-s + 10.0·26-s − 7.41·28-s − 0.450·29-s + 6.98·32-s − 6.25·34-s + 7.38·35-s − 8.02·37-s − 4.04·38-s − 1.41·40-s − 6.42·41-s + 1.07·43-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.744·4-s − 0.662·5-s − 1.88·7-s + 0.337·8-s + 0.874·10-s − 0.530·11-s − 1.48·13-s + 2.48·14-s − 1.19·16-s + 0.812·17-s + 0.497·19-s − 0.492·20-s + 0.700·22-s + 1.51·23-s − 0.561·25-s + 1.96·26-s − 1.40·28-s − 0.0836·29-s + 1.23·32-s − 1.07·34-s + 1.24·35-s − 1.31·37-s − 0.656·38-s − 0.223·40-s − 1.00·41-s + 0.164·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 + 4.98T + 7T^{2} \) |
| 11 | \( 1 + 1.75T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 7.28T + 23T^{2} \) |
| 29 | \( 1 + 0.450T + 29T^{2} \) |
| 37 | \( 1 + 8.02T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 - 1.07T + 43T^{2} \) |
| 47 | \( 1 - 1.10T + 47T^{2} \) |
| 53 | \( 1 + 7.91T + 53T^{2} \) |
| 59 | \( 1 + 6.84T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 1.32T + 71T^{2} \) |
| 73 | \( 1 + 0.185T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 + 0.255T + 97T^{2} \) |
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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
−7.43306097003954292922554431905, −7.09998785823779147822791878513, −6.40308781176883471691019042872, −5.34579279776779989052774573503, −4.71562837352949235870100006225, −3.51120407611861986308377711823, −3.08311201176846880364087316047, −2.10089879771958592354896377140, −0.71913161713639972054970504161, 0,
0.71913161713639972054970504161, 2.10089879771958592354896377140, 3.08311201176846880364087316047, 3.51120407611861986308377711823, 4.71562837352949235870100006225, 5.34579279776779989052774573503, 6.40308781176883471691019042872, 7.09998785823779147822791878513, 7.43306097003954292922554431905
