L(s) = 1 | − 2·3-s − 3·5-s − 4·7-s + 9-s − 6·11-s − 5·13-s + 6·15-s − 2·17-s + 8·21-s − 6·23-s + 4·25-s + 4·27-s − 5·29-s − 8·31-s + 12·33-s + 12·35-s − 6·37-s + 10·39-s − 11·41-s − 2·43-s − 3·45-s + 47-s + 9·49-s + 4·51-s − 10·53-s + 18·55-s + 3·59-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.34·5-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.38·13-s + 1.54·15-s − 0.485·17-s + 1.74·21-s − 1.25·23-s + 4/5·25-s + 0.769·27-s − 0.928·29-s − 1.43·31-s + 2.08·33-s + 2.02·35-s − 0.986·37-s + 1.60·39-s − 1.71·41-s − 0.304·43-s − 0.447·45-s + 0.145·47-s + 9/7·49-s + 0.560·51-s − 1.37·53-s + 2.42·55-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 8069 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.86582080591028, −15.46686264437905, −15.10663326234044, −14.31338039773539, −13.51259761626943, −12.95948107010232, −12.60412392105708, −12.01366628706534, −11.84512005011585, −10.97198204528576, −10.62505229525659, −10.09452176681509, −9.596271227546942, −8.814560014228252, −8.097449800211904, −7.511292041774654, −7.159676922215664, −6.518002299967063, −5.825419397525468, −5.273039738500142, −4.804138105007842, −4.022565008701552, −3.305225793484217, −2.806044090839094, −1.882759535213278, 0, 0, 0,
1.882759535213278, 2.806044090839094, 3.305225793484217, 4.022565008701552, 4.804138105007842, 5.273039738500142, 5.825419397525468, 6.518002299967063, 7.159676922215664, 7.511292041774654, 8.097449800211904, 8.814560014228252, 9.596271227546942, 10.09452176681509, 10.62505229525659, 10.97198204528576, 11.84512005011585, 12.01366628706534, 12.60412392105708, 12.95948107010232, 13.51259761626943, 14.31338039773539, 15.10663326234044, 15.46686264437905, 15.86582080591028