L(s) = 1 | − 0.473·2-s − 1.77·4-s − 3.75·5-s − 7-s + 1.78·8-s + 1.78·10-s + 2.70·13-s + 0.473·14-s + 2.70·16-s − 2.48·17-s − 1.74·19-s + 6.67·20-s − 3.79·23-s + 9.12·25-s − 1.28·26-s + 1.77·28-s + 5.80·29-s + 2.24·31-s − 4.85·32-s + 1.17·34-s + 3.75·35-s − 7.65·37-s + 0.826·38-s − 6.72·40-s + 4.18·41-s + 7.75·43-s + 1.79·46-s + ⋯ |
L(s) = 1 | − 0.334·2-s − 0.887·4-s − 1.68·5-s − 0.377·7-s + 0.632·8-s + 0.563·10-s + 0.750·13-s + 0.126·14-s + 0.675·16-s − 0.602·17-s − 0.400·19-s + 1.49·20-s − 0.792·23-s + 1.82·25-s − 0.251·26-s + 0.335·28-s + 1.07·29-s + 0.403·31-s − 0.858·32-s + 0.201·34-s + 0.635·35-s − 1.25·37-s + 0.134·38-s − 1.06·40-s + 0.653·41-s + 1.18·43-s + 0.265·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3620665469\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3620665469\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.473T + 2T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 - 5.80T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 8.83T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 7.75T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 7.05T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 - 6.99T + 97T^{2} \) |
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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
−8.114066689423162206261271678330, −7.37795847215074781279993094294, −6.60514682667635848107793923858, −5.85157126385971822856153100961, −4.64373845234282289847013390329, −4.40897107732851822060691057819, −3.62227175241455372309276604471, −2.98858466634040221393819382683, −1.46885090353247755508536486113, −0.33586473709459875090234456087,
0.33586473709459875090234456087, 1.46885090353247755508536486113, 2.98858466634040221393819382683, 3.62227175241455372309276604471, 4.40897107732851822060691057819, 4.64373845234282289847013390329, 5.85157126385971822856153100961, 6.60514682667635848107793923858, 7.37795847215074781279993094294, 8.114066689423162206261271678330