L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 11-s − 2·14-s + 16-s − 2·17-s − 6·19-s − 20-s + 22-s + 3·23-s + 25-s − 2·28-s + 29-s + 3·31-s + 32-s − 2·34-s + 2·35-s + 5·37-s − 6·38-s − 40-s + 10·41-s + 5·43-s + 44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s − 0.377·28-s + 0.185·29-s + 0.538·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s + 0.821·37-s − 0.973·38-s − 0.158·40-s + 1.56·41-s + 0.762·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.16294566332243, −15.64343520730474, −15.28474071868143, −14.56412685420895, −14.20921941106627, −13.41047379274564, −12.90406070956679, −12.54057677738038, −12.00202214133328, −11.13063022636068, −10.92267614852346, −10.22289438401822, −9.304994694600401, −9.063343734712837, −8.054052275065476, −7.649943370539517, −6.707531890011835, −6.399802607843896, −5.836316272094596, −4.781453940779551, −4.373690112566228, −3.702872236563880, −2.913069532297228, −2.329894476300594, −1.178744571545509, 0,
1.178744571545509, 2.329894476300594, 2.913069532297228, 3.702872236563880, 4.373690112566228, 4.781453940779551, 5.836316272094596, 6.399802607843896, 6.707531890011835, 7.649943370539517, 8.054052275065476, 9.063343734712837, 9.304994694600401, 10.22289438401822, 10.92267614852346, 11.13063022636068, 12.00202214133328, 12.54057677738038, 12.90406070956679, 13.41047379274564, 14.20921941106627, 14.56412685420895, 15.28474071868143, 15.64343520730474, 16.16294566332243