| L(s) = 1 | − 2.22·2-s + 1.03·3-s + 2.94·4-s + 5-s − 2.30·6-s − 2.02·7-s − 2.09·8-s − 1.92·9-s − 2.22·10-s + 0.0848·11-s + 3.04·12-s + 5.72·13-s + 4.50·14-s + 1.03·15-s − 1.23·16-s − 2.53·17-s + 4.27·18-s + 2.94·20-s − 2.10·21-s − 0.188·22-s − 0.309·23-s − 2.16·24-s + 25-s − 12.7·26-s − 5.10·27-s − 5.95·28-s + 2.62·29-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.598·3-s + 1.47·4-s + 0.447·5-s − 0.941·6-s − 0.765·7-s − 0.739·8-s − 0.641·9-s − 0.702·10-s + 0.0255·11-s + 0.880·12-s + 1.58·13-s + 1.20·14-s + 0.267·15-s − 0.308·16-s − 0.613·17-s + 1.00·18-s + 0.657·20-s − 0.458·21-s − 0.0401·22-s − 0.0644·23-s − 0.442·24-s + 0.200·25-s − 2.49·26-s − 0.982·27-s − 1.12·28-s + 0.487·29-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
| \(L(1)\) |
\(\approx\) |
\(0.9089022752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9089022752\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 - 1.03T + 3T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 - 0.0848T + 11T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 17 | \( 1 + 2.53T + 17T^{2} \) |
| 23 | \( 1 + 0.309T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 - 8.07T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 - 0.650T + 43T^{2} \) |
| 47 | \( 1 - 6.90T + 47T^{2} \) |
| 53 | \( 1 - 14.5T + 53T^{2} \) |
| 59 | \( 1 - 7.47T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 0.115T + 79T^{2} \) |
| 83 | \( 1 + 2.97T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 0.225T + 97T^{2} \) |
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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
−9.067184729018375677453348574011, −8.568618729727056346966494290107, −8.213636861466850775148112649558, −6.96500055197911461144752926475, −6.45749169969085605291276132136, −5.54545522675044972460275730445, −4.00388663650534681658242811996, −2.96204558098204848372736677224, −2.06498671060109601729806598695, −0.797273934588638102980892414329,
0.797273934588638102980892414329, 2.06498671060109601729806598695, 2.96204558098204848372736677224, 4.00388663650534681658242811996, 5.54545522675044972460275730445, 6.45749169969085605291276132136, 6.96500055197911461144752926475, 8.213636861466850775148112649558, 8.568618729727056346966494290107, 9.067184729018375677453348574011
