| L(s) = 1 | + 2.76·2-s − 2.29·3-s + 5.64·4-s + 5-s − 6.33·6-s + 1.54·7-s + 10.0·8-s + 2.25·9-s + 2.76·10-s + 0.891·11-s − 12.9·12-s − 3.93·13-s + 4.26·14-s − 2.29·15-s + 16.5·16-s + 5.53·17-s + 6.22·18-s + 5.64·20-s − 3.53·21-s + 2.46·22-s − 3.08·23-s − 23.0·24-s + 25-s − 10.8·26-s + 1.71·27-s + 8.69·28-s + 3.41·29-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 1.32·3-s + 2.82·4-s + 0.447·5-s − 2.58·6-s + 0.582·7-s + 3.55·8-s + 0.750·9-s + 0.874·10-s + 0.268·11-s − 3.73·12-s − 1.09·13-s + 1.13·14-s − 0.591·15-s + 4.13·16-s + 1.34·17-s + 1.46·18-s + 1.26·20-s − 0.770·21-s + 0.525·22-s − 0.642·23-s − 4.70·24-s + 0.200·25-s − 2.13·26-s + 0.330·27-s + 1.64·28-s + 0.633·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.762109538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.762109538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 + 2.29T + 3T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 - 0.891T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 - 0.0632T + 31T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 5.89T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 + 8.62T + 61T^{2} \) |
| 67 | \( 1 - 0.635T + 67T^{2} \) |
| 71 | \( 1 + 2.40T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 4.46T + 89T^{2} \) |
| 97 | \( 1 + 4.99T + 97T^{2} \) |
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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
−9.720252167453261579267876456466, −8.000861751873028199084072436319, −7.24422863898646010241094420712, −6.40885661257614251087003662913, −5.84820482737007963807681158140, −5.07824315065476960945217374912, −4.76632723643952543536591681161, −3.62684336926666273383550626136, −2.50707681087636164848287312649, −1.36961954188824376902357085898,
1.36961954188824376902357085898, 2.50707681087636164848287312649, 3.62684336926666273383550626136, 4.76632723643952543536591681161, 5.07824315065476960945217374912, 5.84820482737007963807681158140, 6.40885661257614251087003662913, 7.24422863898646010241094420712, 8.000861751873028199084072436319, 9.720252167453261579267876456466
