| L(s) = 1 | − 3·5-s − 7-s − 3·9-s − 3·11-s + 5·13-s − 6·17-s − 4·19-s + 8·23-s + 4·25-s + 8·29-s + 7·31-s + 3·35-s + 2·41-s + 6·43-s + 9·45-s + 6·47-s + 49-s + 53-s + 9·55-s + 9·59-s − 6·61-s + 3·63-s − 15·65-s − 11·67-s − 15·71-s + 2·73-s + 3·77-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 9-s − 0.904·11-s + 1.38·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 4/5·25-s + 1.48·29-s + 1.25·31-s + 0.507·35-s + 0.312·41-s + 0.914·43-s + 1.34·45-s + 0.875·47-s + 1/7·49-s + 0.137·53-s + 1.21·55-s + 1.17·59-s − 0.768·61-s + 0.377·63-s − 1.86·65-s − 1.34·67-s − 1.78·71-s + 0.234·73-s + 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131668988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131668988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.85523850731623, −13.51786067885794, −13.11442692673266, −12.53067846543748, −11.99128390140725, −11.47600630572414, −11.07833728521297, −10.57661668203140, −10.40345593395838, −9.199337581594027, −8.777476574156996, −8.585804220121490, −7.978015298950400, −7.500953144101341, −6.697609061917847, −6.418043270151135, −5.799485635318221, −5.040199938961858, −4.399943743050849, −4.107684922376957, −3.190399324865566, −2.878651243991271, −2.256493535753619, −0.9893656007992680, −0.4234428036073524,
0.4234428036073524, 0.9893656007992680, 2.256493535753619, 2.878651243991271, 3.190399324865566, 4.107684922376957, 4.399943743050849, 5.040199938961858, 5.799485635318221, 6.418043270151135, 6.697609061917847, 7.500953144101341, 7.978015298950400, 8.585804220121490, 8.777476574156996, 9.199337581594027, 10.40345593395838, 10.57661668203140, 11.07833728521297, 11.47600630572414, 11.99128390140725, 12.53067846543748, 13.11442692673266, 13.51786067885794, 13.85523850731623
