L(s) = 1 | + 2-s + 3-s + 4-s + 4.04·5-s + 6-s + 3.45·7-s + 8-s + 9-s + 4.04·10-s + 3.94·11-s + 12-s − 3.02·13-s + 3.45·14-s + 4.04·15-s + 16-s − 17-s + 18-s − 8.28·19-s + 4.04·20-s + 3.45·21-s + 3.94·22-s + 1.98·23-s + 24-s + 11.3·25-s − 3.02·26-s + 27-s + 3.45·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.80·5-s + 0.408·6-s + 1.30·7-s + 0.353·8-s + 0.333·9-s + 1.27·10-s + 1.18·11-s + 0.288·12-s − 0.839·13-s + 0.924·14-s + 1.04·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 1.90·19-s + 0.904·20-s + 0.754·21-s + 0.840·22-s + 0.414·23-s + 0.204·24-s + 2.27·25-s − 0.593·26-s + 0.192·27-s + 0.653·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.821981437\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.821981437\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 19 | \( 1 + 8.28T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 - 1.25T + 29T^{2} \) |
| 31 | \( 1 + 6.55T + 31T^{2} \) |
| 37 | \( 1 - 9.06T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 - 4.40T + 67T^{2} \) |
| 71 | \( 1 + 3.22T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 7.45T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−8.220112991102795189475466273489, −7.14875426261156603667967534778, −6.51011045485228450555943390122, −6.01352383066777897050945942321, −4.91807263756806647572240224256, −4.74586945065266149478986096188, −3.69735668972876813720762139033, −2.54233373152404360131319930829, −1.94931234938579054252242977225, −1.45013660088239179936172210418,
1.45013660088239179936172210418, 1.94931234938579054252242977225, 2.54233373152404360131319930829, 3.69735668972876813720762139033, 4.74586945065266149478986096188, 4.91807263756806647572240224256, 6.01352383066777897050945942321, 6.51011045485228450555943390122, 7.14875426261156603667967534778, 8.220112991102795189475466273489