| L(s) = 1 | + 0.503·2-s + 3.01·3-s − 1.74·4-s + 5-s + 1.51·6-s + 2.48·7-s − 1.88·8-s + 6.06·9-s + 0.503·10-s + 4.58·11-s − 5.25·12-s − 6.06·13-s + 1.25·14-s + 3.01·15-s + 2.54·16-s + 1.83·17-s + 3.05·18-s − 1.74·20-s + 7.49·21-s + 2.30·22-s + 0.125·23-s − 5.67·24-s + 25-s − 3.05·26-s + 9.21·27-s − 4.34·28-s + 3.79·29-s + ⋯ |
| L(s) = 1 | + 0.356·2-s + 1.73·3-s − 0.873·4-s + 0.447·5-s + 0.618·6-s + 0.941·7-s − 0.666·8-s + 2.02·9-s + 0.159·10-s + 1.38·11-s − 1.51·12-s − 1.68·13-s + 0.335·14-s + 0.777·15-s + 0.635·16-s + 0.444·17-s + 0.719·18-s − 0.390·20-s + 1.63·21-s + 0.491·22-s + 0.0262·23-s − 1.15·24-s + 0.200·25-s − 0.598·26-s + 1.77·27-s − 0.821·28-s + 0.703·29-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
| \(L(1)\) |
\(\approx\) |
\(3.834445401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.834445401\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.503T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 23 | \( 1 - 0.125T + 23T^{2} \) |
| 29 | \( 1 - 3.79T + 29T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 - 3.02T + 41T^{2} \) |
| 43 | \( 1 - 8.78T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 9.24T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 5.23T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−9.180205051563280124064543083604, −8.711395443635960283770052460899, −7.73493820231432950282374474391, −7.28461380680789115733568883014, −5.95571742118022438724315656230, −4.81754182054513470165933608313, −4.29097423474741124489936764572, −3.36864809310727922555027811719, −2.41009879622287923320378255507, −1.38301557296156463622950663110,
1.38301557296156463622950663110, 2.41009879622287923320378255507, 3.36864809310727922555027811719, 4.29097423474741124489936764572, 4.81754182054513470165933608313, 5.95571742118022438724315656230, 7.28461380680789115733568883014, 7.73493820231432950282374474391, 8.711395443635960283770052460899, 9.180205051563280124064543083604
