L(s) = 1 | − 3-s + 5-s + 9-s − 11-s + 3·13-s − 15-s + 2·17-s + 2·23-s + 25-s − 27-s + 6·29-s + 11·31-s + 33-s + 4·37-s − 3·39-s − 2·41-s + 11·43-s + 45-s + 6·47-s − 2·51-s − 4·53-s − 55-s − 3·59-s + 8·61-s + 3·65-s + 8·67-s − 2·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.258·15-s + 0.485·17-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.97·31-s + 0.174·33-s + 0.657·37-s − 0.480·39-s − 0.312·41-s + 1.67·43-s + 0.149·45-s + 0.875·47-s − 0.280·51-s − 0.549·53-s − 0.134·55-s − 0.390·59-s + 1.02·61-s + 0.372·65-s + 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.718246853\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.718246853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−15.30771121806007, −14.30474668933189, −13.99722442748596, −13.52784815467244, −12.79652887302702, −12.51475794680956, −11.82899693129229, −11.29805854202794, −10.80263829280962, −10.20678813827291, −9.856484691472407, −9.150103632788531, −8.513398503930268, −8.012273921904018, −7.355201924469848, −6.584020638861820, −6.228903127017151, −5.651089690388246, −4.999507891883741, −4.438503016090979, −3.722053832303292, −2.865273749205553, −2.309697211692543, −1.173140892405636, −0.7702991286837434,
0.7702991286837434, 1.173140892405636, 2.309697211692543, 2.865273749205553, 3.722053832303292, 4.438503016090979, 4.999507891883741, 5.651089690388246, 6.228903127017151, 6.584020638861820, 7.355201924469848, 8.012273921904018, 8.513398503930268, 9.150103632788531, 9.856484691472407, 10.20678813827291, 10.80263829280962, 11.29805854202794, 11.82899693129229, 12.51475794680956, 12.79652887302702, 13.52784815467244, 13.99722442748596, 14.30474668933189, 15.30771121806007