| L(s) = 1 | + 2-s + 0.178·3-s + 4-s − 5-s + 0.178·6-s − 4.63·7-s + 8-s − 2.96·9-s − 10-s + 5.33·11-s + 0.178·12-s − 13-s − 4.63·14-s − 0.178·15-s + 16-s − 6.03·17-s − 2.96·18-s − 0.663·19-s − 20-s − 0.828·21-s + 5.33·22-s − 1.20·23-s + 0.178·24-s + 25-s − 26-s − 1.06·27-s − 4.63·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.103·3-s + 0.5·4-s − 0.447·5-s + 0.0730·6-s − 1.75·7-s + 0.353·8-s − 0.989·9-s − 0.316·10-s + 1.60·11-s + 0.0516·12-s − 0.277·13-s − 1.23·14-s − 0.0461·15-s + 0.250·16-s − 1.46·17-s − 0.699·18-s − 0.152·19-s − 0.223·20-s − 0.180·21-s + 1.13·22-s − 0.251·23-s + 0.0365·24-s + 0.200·25-s − 0.196·26-s − 0.205·27-s − 0.875·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.951111052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.951111052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 0.178T + 3T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 + 0.663T + 19T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 37 | \( 1 - 6.84T + 37T^{2} \) |
| 41 | \( 1 - 7.35T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 0.326T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 3.23T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 8.93T + 83T^{2} \) |
| 89 | \( 1 + 0.384T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−8.627837792188475149174222112801, −7.47216834940710000320720190688, −6.69446324867810952824202706558, −6.30697354892488769666158151920, −5.67758270170868887737788607922, −4.26863760390875415386742308387, −4.02553634262998420332296781101, −2.97556825462200228714434827616, −2.43618260399829584679897347904, −0.68526156478619038567400605932,
0.68526156478619038567400605932, 2.43618260399829584679897347904, 2.97556825462200228714434827616, 4.02553634262998420332296781101, 4.26863760390875415386742308387, 5.67758270170868887737788607922, 6.30697354892488769666158151920, 6.69446324867810952824202706558, 7.47216834940710000320720190688, 8.627837792188475149174222112801
