L(s) = 1 | + 2-s − 3-s + 4-s − 1.63·5-s − 6-s + 7-s + 8-s + 9-s − 1.63·10-s − 6.10·11-s − 12-s + 14-s + 1.63·15-s + 16-s − 3.54·17-s + 18-s − 6.98·19-s − 1.63·20-s − 21-s − 6.10·22-s + 5.07·23-s − 24-s − 2.32·25-s − 27-s + 28-s − 7.10·29-s + 1.63·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.731·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.517·10-s − 1.84·11-s − 0.288·12-s + 0.267·14-s + 0.422·15-s + 0.250·16-s − 0.860·17-s + 0.235·18-s − 1.60·19-s − 0.365·20-s − 0.218·21-s − 1.30·22-s + 1.05·23-s − 0.204·24-s − 0.465·25-s − 0.192·27-s + 0.188·28-s − 1.31·29-s + 0.298·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.392510865\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.392510865\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 1.63T + 5T^{2} \) |
| 11 | \( 1 + 6.10T + 11T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 + 6.98T + 19T^{2} \) |
| 23 | \( 1 - 5.07T + 23T^{2} \) |
| 29 | \( 1 + 7.10T + 29T^{2} \) |
| 31 | \( 1 - 5.14T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 + 4.12T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 1.33T + 59T^{2} \) |
| 61 | \( 1 - 9.77T + 61T^{2} \) |
| 67 | \( 1 - 9.30T + 67T^{2} \) |
| 71 | \( 1 + 6.97T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 - 0.425T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 5.43T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−7.78350134179147322361823760741, −7.27305743596340259640261012114, −6.38434238990300634959892107166, −5.77568931198585271944228157830, −4.95125768193290168100736898344, −4.51161229496770260493688676114, −3.79461614844392518123725533236, −2.68010194846869781894402706162, −2.07846929920352107143716410788, −0.52373555138716405879647323749,
0.52373555138716405879647323749, 2.07846929920352107143716410788, 2.68010194846869781894402706162, 3.79461614844392518123725533236, 4.51161229496770260493688676114, 4.95125768193290168100736898344, 5.77568931198585271944228157830, 6.38434238990300634959892107166, 7.27305743596340259640261012114, 7.78350134179147322361823760741