| L(s) = 1 | − 5.58·2-s + 1.54·3-s + 23.1·4-s + 18.5·5-s − 8.62·6-s + 4.56·7-s − 84.6·8-s − 24.6·9-s − 103.·10-s + 35.8·12-s − 29.4·13-s − 25.4·14-s + 28.5·15-s + 287.·16-s + 17·17-s + 137.·18-s − 138.·19-s + 428.·20-s + 7.05·21-s − 13.3·23-s − 130.·24-s + 217.·25-s + 164.·26-s − 79.7·27-s + 105.·28-s + 159.·29-s − 159.·30-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.297·3-s + 2.89·4-s + 1.65·5-s − 0.587·6-s + 0.246·7-s − 3.74·8-s − 0.911·9-s − 3.26·10-s + 0.861·12-s − 0.628·13-s − 0.486·14-s + 0.492·15-s + 4.49·16-s + 0.242·17-s + 1.79·18-s − 1.67·19-s + 4.79·20-s + 0.0732·21-s − 0.121·23-s − 1.11·24-s + 1.73·25-s + 1.23·26-s − 0.568·27-s + 0.713·28-s + 1.01·29-s − 0.971·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.071192199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.071192199\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 5.58T + 8T^{2} \) |
| 3 | \( 1 - 1.54T + 27T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 7 | \( 1 - 4.56T + 343T^{2} \) |
| 13 | \( 1 + 29.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 91.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 590.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 170.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 565.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 504.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 859.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.90e3T + 9.12e5T^{2} \) |
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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
−8.872274475964119454049493915450, −8.307919553517975387761434040904, −7.48291844137311129455012058264, −6.42717363726054484044864578989, −6.13264690215345408477160148390, −5.12602570355234174578995659118, −3.15661453366156920013950207932, −2.22305418925813953562393243170, −1.90433196941314760076596613834, −0.60089195010812185973329463405,
0.60089195010812185973329463405, 1.90433196941314760076596613834, 2.22305418925813953562393243170, 3.15661453366156920013950207932, 5.12602570355234174578995659118, 6.13264690215345408477160148390, 6.42717363726054484044864578989, 7.48291844137311129455012058264, 8.307919553517975387761434040904, 8.872274475964119454049493915450
