| L(s) = 1 | + 2-s − 2.23·3-s + 4-s + 5-s − 2.23·6-s + 3.23·7-s + 8-s + 2.00·9-s + 10-s − 0.236·11-s − 2.23·12-s − 3.23·13-s + 3.23·14-s − 2.23·15-s + 16-s + 4.47·17-s + 2.00·18-s + 4.23·19-s + 20-s − 7.23·21-s − 0.236·22-s − 1.76·23-s − 2.23·24-s − 4·25-s − 3.23·26-s + 2.23·27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s − 0.912·6-s + 1.22·7-s + 0.353·8-s + 0.666·9-s + 0.316·10-s − 0.0711·11-s − 0.645·12-s − 0.897·13-s + 0.864·14-s − 0.577·15-s + 0.250·16-s + 1.08·17-s + 0.471·18-s + 0.971·19-s + 0.223·20-s − 1.57·21-s − 0.0503·22-s − 0.367·23-s − 0.456·24-s − 0.800·25-s − 0.634·26-s + 0.430·27-s + 0.611·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.934419142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934419142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 + 6.47T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
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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
−10.32481702134480335745756151991, −9.723605817208830015907629702348, −8.204352759652178116214598202661, −7.44953280414046437937874287737, −6.44116444032137020041614035301, −5.40668935333792939185731574652, −5.22964447690082351168695655388, −4.15617552990097400756556311261, −2.57529465246915388061748919956, −1.17917255355302963741113192973,
1.17917255355302963741113192973, 2.57529465246915388061748919956, 4.15617552990097400756556311261, 5.22964447690082351168695655388, 5.40668935333792939185731574652, 6.44116444032137020041614035301, 7.44953280414046437937874287737, 8.204352759652178116214598202661, 9.723605817208830015907629702348, 10.32481702134480335745756151991
