| L(s) = 1 | − 2-s + 0.988·3-s + 4-s + 4.28·5-s − 0.988·6-s + 1.82·7-s − 8-s − 2.02·9-s − 4.28·10-s − 0.346·11-s + 0.988·12-s + 0.174·13-s − 1.82·14-s + 4.23·15-s + 16-s + 7.56·17-s + 2.02·18-s + 4.50·19-s + 4.28·20-s + 1.80·21-s + 0.346·22-s − 7.00·23-s − 0.988·24-s + 13.3·25-s − 0.174·26-s − 4.96·27-s + 1.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.570·3-s + 0.5·4-s + 1.91·5-s − 0.403·6-s + 0.689·7-s − 0.353·8-s − 0.674·9-s − 1.35·10-s − 0.104·11-s + 0.285·12-s + 0.0483·13-s − 0.487·14-s + 1.09·15-s + 0.250·16-s + 1.83·17-s + 0.476·18-s + 1.03·19-s + 0.958·20-s + 0.393·21-s + 0.0739·22-s − 1.46·23-s − 0.201·24-s + 2.67·25-s − 0.0341·26-s − 0.955·27-s + 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.790183214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.790183214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 0.988T + 3T^{2} \) |
| 5 | \( 1 - 4.28T + 5T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 0.346T + 11T^{2} \) |
| 13 | \( 1 - 0.174T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 + 7.00T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 - 4.45T + 53T^{2} \) |
| 59 | \( 1 - 0.131T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 0.00583T + 71T^{2} \) |
| 73 | \( 1 - 6.71T + 73T^{2} \) |
| 79 | \( 1 - 3.84T + 79T^{2} \) |
| 83 | \( 1 - 6.40T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−8.454883453837869669480316087011, −7.970423516107607307808858120614, −7.11747149218534545812669326981, −6.11102247839002393724417738418, −5.61240429195088783845913656562, −5.03504457988446160548655714618, −3.45584665787256670437965198721, −2.68801160838190010271618652634, −1.89106188817758680877160721640, −1.13103167823282889972350738106,
1.13103167823282889972350738106, 1.89106188817758680877160721640, 2.68801160838190010271618652634, 3.45584665787256670437965198721, 5.03504457988446160548655714618, 5.61240429195088783845913656562, 6.11102247839002393724417738418, 7.11747149218534545812669326981, 7.970423516107607307808858120614, 8.454883453837869669480316087011
