| L(s) = 1 | − 3.23i·3-s − 1.23i·5-s + 7-s − 7.47·9-s − 2.47i·11-s − 5.23i·13-s − 4.00·15-s − 4.47·17-s + 3.23i·19-s − 3.23i·21-s + 4·23-s + 3.47·25-s + 14.4i·27-s − 4.47i·29-s − 6.47·31-s + ⋯ |
| L(s) = 1 | − 1.86i·3-s − 0.552i·5-s + 0.377·7-s − 2.49·9-s − 0.745i·11-s − 1.45i·13-s − 1.03·15-s − 1.08·17-s + 0.742i·19-s − 0.706i·21-s + 0.834·23-s + 0.694·25-s + 2.78i·27-s − 0.830i·29-s − 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165786766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165786766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 3.23iT - 3T^{2} \) |
| 5 | \( 1 + 1.23iT - 5T^{2} \) |
| 11 | \( 1 + 2.47iT - 11T^{2} \) |
| 13 | \( 1 + 5.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 3.23iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 - 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4.76iT - 59T^{2} \) |
| 61 | \( 1 + 6.76iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 4.76iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−8.371669883784478328364598629495, −8.127465573294815863302700091700, −7.26052536672154890241563840156, −6.45416955985421466982142549337, −5.70530137697215057746217609566, −4.99319201518436424976142896295, −3.40647380099125503466685598526, −2.44797108055215394285968784261, −1.36475606281304590413121895929, −0.43798018281833741942547548321,
2.14317264395269263045244909411, 3.16021453384950258945363346128, 4.19434586822294858663568145310, 4.64971868371406543862829813315, 5.42345528650131709159111661812, 6.63424138849632368611417493316, 7.25888942354434239419142579481, 8.716110638785030528178930940772, 9.075687562106864817989857348191, 9.644505344708166245993344081943
