| L(s) = 1 | − 0.834·2-s − 1.30·4-s + 2.83·5-s + 4.22·7-s + 2.75·8-s − 2.36·10-s − 0.137·11-s − 3.68·13-s − 3.52·14-s + 0.302·16-s + 6.13·17-s − 5.27·19-s − 3.69·20-s + 0.115·22-s + 5.39·23-s + 3.03·25-s + 3.07·26-s − 5.50·28-s − 4.83·29-s − 7.51·31-s − 5.76·32-s − 5.12·34-s + 11.9·35-s + 7.42·37-s + 4.40·38-s + 7.81·40-s + 11.5·41-s + ⋯ |
| L(s) = 1 | − 0.590·2-s − 0.651·4-s + 1.26·5-s + 1.59·7-s + 0.975·8-s − 0.748·10-s − 0.0415·11-s − 1.02·13-s − 0.942·14-s + 0.0756·16-s + 1.48·17-s − 1.21·19-s − 0.825·20-s + 0.0245·22-s + 1.12·23-s + 0.607·25-s + 0.602·26-s − 1.04·28-s − 0.897·29-s − 1.34·31-s − 1.01·32-s − 0.878·34-s + 2.02·35-s + 1.22·37-s + 0.714·38-s + 1.23·40-s + 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.370922322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370922322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 0.834T + 2T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 + 0.137T + 11T^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 - 6.13T + 17T^{2} \) |
| 19 | \( 1 + 5.27T + 19T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 5.88T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 + 9.67T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 4.01T + 67T^{2} \) |
| 73 | \( 1 - 7.84T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 + 9.81T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−10.59726638062448355661397777147, −9.409540847222424142608467908103, −9.175350595555873842716820995525, −7.894299784957261648450401831138, −7.47735614875614802157921903263, −5.80534321398579984237352239925, −5.15801363493190558245683142483, −4.25456018048159361621935175662, −2.32352222819263324372860664943, −1.26574902734765406863323212060,
1.26574902734765406863323212060, 2.32352222819263324372860664943, 4.25456018048159361621935175662, 5.15801363493190558245683142483, 5.80534321398579984237352239925, 7.47735614875614802157921903263, 7.894299784957261648450401831138, 9.175350595555873842716820995525, 9.409540847222424142608467908103, 10.59726638062448355661397777147
