L(s) = 1 | − 0.477·3-s + i·5-s + (−2.09 − 1.61i)7-s − 2.77·9-s + 4.66i·11-s − 3.77i·13-s − 0.477i·15-s − 3.77i·17-s + 7.42·19-s + (1 + 0.772i)21-s − 3.23i·23-s − 25-s + 2.75·27-s − 3.77·29-s + 0.954·31-s + ⋯ |
L(s) = 1 | − 0.275·3-s + 0.447i·5-s + (−0.791 − 0.611i)7-s − 0.924·9-s + 1.40i·11-s − 1.04i·13-s − 0.123i·15-s − 0.914i·17-s + 1.70·19-s + (0.218 + 0.168i)21-s − 0.674i·23-s − 0.200·25-s + 0.530·27-s − 0.700·29-s + 0.171·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.084346852\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.084346852\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.09 + 1.61i)T \) |
good | 3 | \( 1 + 0.477T + 3T^{2} \) |
| 11 | \( 1 - 4.66iT - 11T^{2} \) |
| 13 | \( 1 + 3.77iT - 13T^{2} \) |
| 17 | \( 1 + 3.77iT - 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 + 3.23iT - 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 - 0.954T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 12.5iT - 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 9.33T + 59T^{2} \) |
| 61 | \( 1 - 13.5iT - 61T^{2} \) |
| 67 | \( 1 - 6.09iT - 67T^{2} \) |
| 71 | \( 1 - 9.33iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 11.1iT - 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 8.22iT - 97T^{2} \) |
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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
−9.380626340157480996672487722349, −8.299065523350994108822465904407, −7.25917426807853307843555112639, −7.11635193154381845556090047063, −5.94591507008299075234805297889, −5.30018602588740853895065569114, −4.32468607479509648421597680549, −3.16729701132413012336726757640, −2.63919839948491399032294876382, −0.892190898119500746018280046689,
0.51172360528222489023436305126, 2.00837232905963280813645540206, 3.26490805730872388603250200038, 3.78570831745372803773432254012, 5.43409808847021982297101470093, 5.53452015277216702060625132321, 6.42664744874796473523356227025, 7.33071735917224267823988068649, 8.424847964826703327325387392621, 8.908528358394159980229125854268