| L(s) = 1 | + 2.46·2-s + 4.05·4-s − 0.989·5-s − 4.32·7-s + 5.05·8-s − 2.43·10-s − 3.61·13-s − 10.6·14-s + 4.32·16-s + 2.64·17-s + 3.82·19-s − 4.01·20-s − 3.45·23-s − 4.02·25-s − 8.90·26-s − 17.5·28-s − 10.4·29-s − 1.68·31-s + 0.530·32-s + 6.49·34-s + 4.27·35-s + 3.23·37-s + 9.41·38-s − 4.99·40-s − 11.9·41-s + 2.12·43-s − 8.48·46-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.02·4-s − 0.442·5-s − 1.63·7-s + 1.78·8-s − 0.770·10-s − 1.00·13-s − 2.84·14-s + 1.08·16-s + 0.640·17-s + 0.878·19-s − 0.897·20-s − 0.719·23-s − 0.804·25-s − 1.74·26-s − 3.31·28-s − 1.93·29-s − 0.303·31-s + 0.0938·32-s + 1.11·34-s + 0.723·35-s + 0.532·37-s + 1.52·38-s − 0.790·40-s − 1.85·41-s + 0.323·43-s − 1.25·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 + 0.989T + 5T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.68T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 4.81T + 61T^{2} \) |
| 67 | \( 1 - 0.854T + 67T^{2} \) |
| 71 | \( 1 - 3.77T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 6.14T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 0.371T + 97T^{2} \) |
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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
−7.83335500762773034168967009979, −7.24732254124379567802892139683, −6.61496793502611444640305690751, −5.74081677833267572692929905436, −5.34032731467807632608962212363, −4.20960615043554006363174556626, −3.55933815025587070948246983403, −3.04990253267824224361840593522, −2.02368594789201115214186035171, 0,
2.02368594789201115214186035171, 3.04990253267824224361840593522, 3.55933815025587070948246983403, 4.20960615043554006363174556626, 5.34032731467807632608962212363, 5.74081677833267572692929905436, 6.61496793502611444640305690751, 7.24732254124379567802892139683, 7.83335500762773034168967009979
