| L(s) = 1 | − 2·5-s + 3·7-s + 5·11-s − 3·13-s − 3·17-s − 5·19-s + 3·23-s − 25-s − 4·31-s − 6·35-s + 37-s − 6·41-s − 4·43-s − 4·47-s + 2·49-s + 3·53-s − 10·55-s + 14·59-s − 14·61-s + 6·65-s − 12·67-s + 12·71-s + 13·73-s + 15·77-s − 6·79-s − 7·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.13·7-s + 1.50·11-s − 0.832·13-s − 0.727·17-s − 1.14·19-s + 0.625·23-s − 1/5·25-s − 0.718·31-s − 1.01·35-s + 0.164·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s + 2/7·49-s + 0.412·53-s − 1.34·55-s + 1.82·59-s − 1.79·61-s + 0.744·65-s − 1.46·67-s + 1.42·71-s + 1.52·73-s + 1.70·77-s − 0.675·79-s − 0.768·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.967713642061687623421053603131, −6.99736747141541711671761771102, −6.69937524878685284649889130536, −5.56541402688287257458755341938, −4.65910267030752847333791716575, −4.24643410982889396927426767025, −3.46801459541777731548305072088, −2.22301988841338891265188347300, −1.41329202412733889384814809600, 0,
1.41329202412733889384814809600, 2.22301988841338891265188347300, 3.46801459541777731548305072088, 4.24643410982889396927426767025, 4.65910267030752847333791716575, 5.56541402688287257458755341938, 6.69937524878685284649889130536, 6.99736747141541711671761771102, 7.967713642061687623421053603131
