| L(s) = 1 | + 5·5-s − 24.7·7-s − 8.16·11-s − 46.7·13-s − 60.3·17-s + 111.·19-s + 36.9·23-s + 25·25-s − 33.2·29-s + 124.·31-s − 123.·35-s − 438.·37-s − 508.·41-s + 48.5·43-s − 248.·47-s + 271.·49-s + 320.·53-s − 40.8·55-s − 652.·59-s + 693.·61-s − 233.·65-s + 12.0·67-s + 1.16e3·71-s − 122.·73-s + 202.·77-s + 441.·79-s + 428.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.33·7-s − 0.223·11-s − 0.997·13-s − 0.860·17-s + 1.34·19-s + 0.334·23-s + 0.200·25-s − 0.212·29-s + 0.720·31-s − 0.598·35-s − 1.94·37-s − 1.93·41-s + 0.172·43-s − 0.769·47-s + 0.791·49-s + 0.830·53-s − 0.100·55-s − 1.43·59-s + 1.45·61-s − 0.446·65-s + 0.0219·67-s + 1.95·71-s − 0.195·73-s + 0.299·77-s + 0.629·79-s + 0.566·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.252228083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252228083\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 24.7T + 343T^{2} \) |
| 11 | \( 1 + 8.16T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 33.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 438.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 508.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 48.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 248.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 320.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 12.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 122.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 441.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 428.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 500.T + 9.12e5T^{2} \) |
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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
−8.911657580914181102711911098388, −7.920816652842604096063382956568, −6.84567770276326696293066134168, −6.67121043329780711788790803197, −5.45218243618999302573208374452, −4.91461584590121560834030244784, −3.59049683057299957096515029460, −2.91938802305666395474060611604, −1.92487551084706661762631249280, −0.48418247747877377240088489730,
0.48418247747877377240088489730, 1.92487551084706661762631249280, 2.91938802305666395474060611604, 3.59049683057299957096515029460, 4.91461584590121560834030244784, 5.45218243618999302573208374452, 6.67121043329780711788790803197, 6.84567770276326696293066134168, 7.920816652842604096063382956568, 8.911657580914181102711911098388
