| L(s) = 1 | − 1.39·2-s − 0.0671·4-s + 5-s − 1.27·7-s + 2.87·8-s − 1.39·10-s + 1.41·13-s + 1.77·14-s − 3.86·16-s + 5·17-s − 0.158·19-s − 0.0671·20-s + 5.00·23-s + 25-s − 1.97·26-s + 0.0858·28-s + 6.27·29-s + 3.04·31-s − 0.379·32-s − 6.95·34-s − 1.27·35-s − 4.69·37-s + 0.219·38-s + 2.87·40-s + 7.58·41-s − 5.41·43-s − 6.96·46-s + ⋯ |
| L(s) = 1 | − 0.983·2-s − 0.0335·4-s + 0.447·5-s − 0.482·7-s + 1.01·8-s − 0.439·10-s + 0.393·13-s + 0.474·14-s − 0.965·16-s + 1.21·17-s − 0.0362·19-s − 0.0150·20-s + 1.04·23-s + 0.200·25-s − 0.386·26-s + 0.0162·28-s + 1.16·29-s + 0.547·31-s − 0.0671·32-s − 1.19·34-s − 0.215·35-s − 0.772·37-s + 0.0356·38-s + 0.454·40-s + 1.18·41-s − 0.825·43-s − 1.02·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171803849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171803849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 0.158T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 - 8.23T + 47T^{2} \) |
| 53 | \( 1 + 9.36T + 53T^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 - 6.77T + 71T^{2} \) |
| 73 | \( 1 + 8.66T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 6.41T + 97T^{2} \) |
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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
−8.257659545271910758180566970704, −7.63896559356954784424655544894, −6.85420323822915681401676439485, −6.17651270856413147270229537838, −5.27707841718082109963558048100, −4.59751559266426748048419129339, −3.55601055171882857246969098540, −2.73653387617468068130816705969, −1.51529117426379229183103713603, −0.73727427499190593799184026713,
0.73727427499190593799184026713, 1.51529117426379229183103713603, 2.73653387617468068130816705969, 3.55601055171882857246969098540, 4.59751559266426748048419129339, 5.27707841718082109963558048100, 6.17651270856413147270229537838, 6.85420323822915681401676439485, 7.63896559356954784424655544894, 8.257659545271910758180566970704
