| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 4·11-s + 2·13-s + 4·14-s + 16-s + 2·17-s − 20-s + 4·22-s + 8·23-s + 25-s + 2·26-s + 4·28-s + 6·29-s − 4·31-s + 32-s + 2·34-s − 4·35-s + 10·37-s − 40-s − 2·41-s + 12·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.676·35-s + 1.64·37-s − 0.158·40-s − 0.312·41-s + 1.82·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.359868820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.359868820\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.92768274161012, −14.61996766480620, −14.02241241881262, −13.62084347512263, −12.92735432427644, −12.23668739531539, −11.96277465212936, −11.30892774171584, −10.95869266277856, −10.61860709822272, −9.574235202776666, −9.006324952981799, −8.539033361607647, −7.766116431511250, −7.505429711277855, −6.685160029643655, −6.214259402687161, −5.387314674760514, −4.946443023933806, −4.224587655386284, −3.927084798586643, −3.033777153352932, −2.347005191409628, −1.219833926327512, −1.093231570345029,
1.093231570345029, 1.219833926327512, 2.347005191409628, 3.033777153352932, 3.927084798586643, 4.224587655386284, 4.946443023933806, 5.387314674760514, 6.214259402687161, 6.685160029643655, 7.505429711277855, 7.766116431511250, 8.539033361607647, 9.006324952981799, 9.574235202776666, 10.61860709822272, 10.95869266277856, 11.30892774171584, 11.96277465212936, 12.23668739531539, 12.92735432427644, 13.62084347512263, 14.02241241881262, 14.61996766480620, 14.92768274161012
