| L(s) = 1 | − 1.30·3-s − 2.30·5-s − 1.30·9-s + 4·11-s − 2.60·13-s + 3·15-s + 17-s − 1.39·19-s + 4·23-s + 0.302·25-s + 5.60·27-s − 5.21·29-s − 3.69·31-s − 5.21·33-s − 11.8·37-s + 3.39·39-s − 6.51·41-s − 0.697·43-s + 3.00·45-s − 4.60·47-s − 1.30·51-s + 4.30·53-s − 9.21·55-s + 1.81·57-s + 8·59-s + 2.51·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.752·3-s − 1.02·5-s − 0.434·9-s + 1.20·11-s − 0.722·13-s + 0.774·15-s + 0.242·17-s − 0.319·19-s + 0.834·23-s + 0.0605·25-s + 1.07·27-s − 0.967·29-s − 0.664·31-s − 0.907·33-s − 1.94·37-s + 0.543·39-s − 1.01·41-s − 0.106·43-s + 0.447·45-s − 0.671·47-s − 0.182·51-s + 0.591·53-s − 1.24·55-s + 0.240·57-s + 1.04·59-s + 0.321·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7315443924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7315443924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 + 0.697T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 4.30T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 4.60T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
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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
−8.671585257688785724370683754813, −7.82180971822947860581144124178, −7.00439962645016534362261510177, −6.54119028399847261641829827437, −5.46343236223243984320269451537, −4.92662995774910069955976555214, −3.87696307826549989272217908119, −3.32039345517112502669803111116, −1.90459434540108135576791933344, −0.51938492046778447778210551007,
0.51938492046778447778210551007, 1.90459434540108135576791933344, 3.32039345517112502669803111116, 3.87696307826549989272217908119, 4.92662995774910069955976555214, 5.46343236223243984320269451537, 6.54119028399847261641829827437, 7.00439962645016534362261510177, 7.82180971822947860581144124178, 8.671585257688785724370683754813
