| L(s) = 1 | − 2-s − 0.717·3-s + 4-s + 3.51·5-s + 0.717·6-s − 3.26·7-s − 8-s − 2.48·9-s − 3.51·10-s + 4.06·11-s − 0.717·12-s − 4.30·13-s + 3.26·14-s − 2.52·15-s + 16-s − 5.32·17-s + 2.48·18-s + 1.17·19-s + 3.51·20-s + 2.34·21-s − 4.06·22-s − 5.52·23-s + 0.717·24-s + 7.37·25-s + 4.30·26-s + 3.93·27-s − 3.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.414·3-s + 0.5·4-s + 1.57·5-s + 0.292·6-s − 1.23·7-s − 0.353·8-s − 0.828·9-s − 1.11·10-s + 1.22·11-s − 0.207·12-s − 1.19·13-s + 0.873·14-s − 0.651·15-s + 0.250·16-s − 1.29·17-s + 0.585·18-s + 0.269·19-s + 0.786·20-s + 0.511·21-s − 0.867·22-s − 1.15·23-s + 0.146·24-s + 1.47·25-s + 0.843·26-s + 0.757·27-s − 0.617·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9475625019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9475625019\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 0.717T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 0.940T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 7.96T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 + 8.78T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
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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
−8.229588545862456773787648933154, −7.16312042320573795411851440596, −6.40156753118786861450173369226, −6.26861229930008162129001306696, −5.53462765556080747247434453207, −4.57691381499090739911570266811, −3.40132196269891883416776689722, −2.47328618591983215894915674372, −1.92797868184190997293445394583, −0.55231826651036126550106645644,
0.55231826651036126550106645644, 1.92797868184190997293445394583, 2.47328618591983215894915674372, 3.40132196269891883416776689722, 4.57691381499090739911570266811, 5.53462765556080747247434453207, 6.26861229930008162129001306696, 6.40156753118786861450173369226, 7.16312042320573795411851440596, 8.229588545862456773787648933154
