| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.23i·5-s + 0.411i·7-s + 2.82·8-s − 3.16i·10-s + 6.98i·11-s − 7.65·13-s + 0.581i·14-s + 4.00·16-s + 22.0i·17-s − 8.31i·19-s − 4.47i·20-s + 9.88i·22-s + (−17.3 + 15.0i)23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.447i·5-s + 0.0587i·7-s + 0.353·8-s − 0.316i·10-s + 0.635i·11-s − 0.589·13-s + 0.0415i·14-s + 0.250·16-s + 1.29i·17-s − 0.437i·19-s − 0.223i·20-s + 0.449i·22-s + (−0.755 + 0.655i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.276964309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276964309\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (17.3 - 15.0i)T \) |
| good | 7 | \( 1 - 0.411iT - 49T^{2} \) |
| 11 | \( 1 - 6.98iT - 121T^{2} \) |
| 13 | \( 1 + 7.65T + 169T^{2} \) |
| 17 | \( 1 - 22.0iT - 289T^{2} \) |
| 19 | \( 1 + 8.31iT - 361T^{2} \) |
| 29 | \( 1 + 32.6T + 841T^{2} \) |
| 31 | \( 1 + 17.7T + 961T^{2} \) |
| 37 | \( 1 + 22.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 7.58iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 39.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 70.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 26.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 62.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 35.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 143. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 95.9iT - 9.40e3T^{2} \) |
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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
−9.267574589293668273488680967114, −8.361450802946036768145620393179, −7.54318328368302678908542330128, −6.87366469132166981224586996471, −5.82591600316753629124907867367, −5.28163318154631517464621379918, −4.28083547698716552407356550776, −3.67751467997013037348676063415, −2.36745152762608037026346659113, −1.53465871842269054873790829658,
0.22163258756603533787066829302, 1.85369379107182327449735994471, 2.86233913212061649424309544458, 3.61794412297937624915123396293, 4.62588562712301012184653743235, 5.43634493549360049171345647099, 6.21296415299328507674075238646, 7.04167684997701631898638409369, 7.67156990899961449895665600316, 8.571729276249804330864173129989
