| L(s) = 1 | − 1.87·2-s − 1.77·3-s + 1.51·4-s + 5-s + 3.33·6-s + 4.25·7-s + 0.901·8-s + 0.159·9-s − 1.87·10-s − 2.70·12-s + 1.36·13-s − 7.98·14-s − 1.77·15-s − 4.73·16-s + 2.09·17-s − 0.299·18-s + 0.604·19-s + 1.51·20-s − 7.56·21-s − 4.39·23-s − 1.60·24-s + 25-s − 2.55·26-s + 5.04·27-s + 6.46·28-s − 6.63·29-s + 3.33·30-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 1.02·3-s + 0.759·4-s + 0.447·5-s + 1.36·6-s + 1.60·7-s + 0.318·8-s + 0.0531·9-s − 0.593·10-s − 0.779·12-s + 0.377·13-s − 2.13·14-s − 0.458·15-s − 1.18·16-s + 0.508·17-s − 0.0705·18-s + 0.138·19-s + 0.339·20-s − 1.65·21-s − 0.916·23-s − 0.327·24-s + 0.200·25-s − 0.500·26-s + 0.971·27-s + 1.22·28-s − 1.23·29-s + 0.608·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6447420072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6447420072\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 - 0.604T + 19T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 + 6.63T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 - 7.40T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 - 9.86T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 + 0.722T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 0.952T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−10.81923980747304794773415624293, −9.789450759579999885227731640820, −8.941778165051189525738264100272, −8.030009234004533529306819749843, −7.44807360774094599999828725945, −6.10107983838590632936608903311, −5.33322061232929930404373761811, −4.30527704672202159909961329509, −2.08303021296616042965844033710, −0.948431305586601174813472817241,
0.948431305586601174813472817241, 2.08303021296616042965844033710, 4.30527704672202159909961329509, 5.33322061232929930404373761811, 6.10107983838590632936608903311, 7.44807360774094599999828725945, 8.030009234004533529306819749843, 8.941778165051189525738264100272, 9.789450759579999885227731640820, 10.81923980747304794773415624293
