| L(s) = 1 | + 2-s + 3.39·3-s + 4-s + 3.61·5-s + 3.39·6-s − 0.821·7-s + 8-s + 8.51·9-s + 3.61·10-s − 2.67·11-s + 3.39·12-s + 3.43·13-s − 0.821·14-s + 12.2·15-s + 16-s − 5.28·17-s + 8.51·18-s − 19-s + 3.61·20-s − 2.78·21-s − 2.67·22-s − 6.30·23-s + 3.39·24-s + 8.09·25-s + 3.43·26-s + 18.7·27-s − 0.821·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.95·3-s + 0.5·4-s + 1.61·5-s + 1.38·6-s − 0.310·7-s + 0.353·8-s + 2.83·9-s + 1.14·10-s − 0.807·11-s + 0.979·12-s + 0.953·13-s − 0.219·14-s + 3.17·15-s + 0.250·16-s − 1.28·17-s + 2.00·18-s − 0.229·19-s + 0.809·20-s − 0.608·21-s − 0.571·22-s − 1.31·23-s + 0.692·24-s + 1.61·25-s + 0.674·26-s + 3.60·27-s − 0.155·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.434666624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.434666624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 + 0.821T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 - 1.96T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 9.91T + 61T^{2} \) |
| 67 | \( 1 + 9.04T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−7.976217958109292908287246148670, −7.07248193926247400105736169965, −6.34665208767504874112141405184, −5.98344898945205159867450741885, −4.67061763197780637884237971232, −4.36231090197619770317009623439, −3.14849973555261524270187821741, −2.80830411387348317483027476128, −2.03477180091774373592725536646, −1.48238758032561397707754680718,
1.48238758032561397707754680718, 2.03477180091774373592725536646, 2.80830411387348317483027476128, 3.14849973555261524270187821741, 4.36231090197619770317009623439, 4.67061763197780637884237971232, 5.98344898945205159867450741885, 6.34665208767504874112141405184, 7.07248193926247400105736169965, 7.976217958109292908287246148670
