| L(s) = 1 | − 2.50·2-s − 0.513·3-s + 4.27·4-s + 1.36·5-s + 1.28·6-s − 1.89·7-s − 5.69·8-s − 2.73·9-s − 3.42·10-s + 4.51·11-s − 2.19·12-s − 1.82·13-s + 4.75·14-s − 0.701·15-s + 5.71·16-s − 2.06·17-s + 6.85·18-s − 19-s + 5.84·20-s + 0.974·21-s − 11.3·22-s + 8.30·23-s + 2.92·24-s − 3.13·25-s + 4.57·26-s + 2.94·27-s − 8.11·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.296·3-s + 2.13·4-s + 0.611·5-s + 0.524·6-s − 0.717·7-s − 2.01·8-s − 0.912·9-s − 1.08·10-s + 1.36·11-s − 0.633·12-s − 0.506·13-s + 1.27·14-s − 0.181·15-s + 1.42·16-s − 0.500·17-s + 1.61·18-s − 0.229·19-s + 1.30·20-s + 0.212·21-s − 2.41·22-s + 1.73·23-s + 0.596·24-s − 0.626·25-s + 0.897·26-s + 0.566·27-s − 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6002095967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6002095967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 + 0.513T + 3T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 23 | \( 1 - 8.30T + 23T^{2} \) |
| 29 | \( 1 - 0.0175T + 29T^{2} \) |
| 31 | \( 1 - 6.71T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 - 9.28T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 8.40T + 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 + 0.166T + 61T^{2} \) |
| 67 | \( 1 + 0.834T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 + 7.32T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 9.28T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.433114918370714498373936677298, −7.23596974522271488060034137557, −6.91369379127159289385674957290, −6.20655389359235926292485602896, −5.64574272952921885042877665593, −4.46634548281489541325649488936, −3.20441307955475861081017260801, −2.50273951971888802658103860427, −1.53367265094080859384666676712, −0.55296401698751159331284076638,
0.55296401698751159331284076638, 1.53367265094080859384666676712, 2.50273951971888802658103860427, 3.20441307955475861081017260801, 4.46634548281489541325649488936, 5.64574272952921885042877665593, 6.20655389359235926292485602896, 6.91369379127159289385674957290, 7.23596974522271488060034137557, 8.433114918370714498373936677298
