| L(s) = 1 | + 1.35·2-s − 0.151·4-s − 1.15·5-s − 1.76·7-s − 2.92·8-s − 1.56·10-s + 11-s − 2.03·13-s − 2.39·14-s − 3.67·16-s + 4.55·17-s + 7.04·19-s + 0.174·20-s + 1.35·22-s − 2.67·23-s − 3.67·25-s − 2.76·26-s + 0.266·28-s − 6.08·29-s + 8.49·31-s + 0.854·32-s + 6.19·34-s + 2.02·35-s + 4.43·37-s + 9.57·38-s + 3.36·40-s + 10.6·41-s + ⋯ |
| L(s) = 1 | + 0.961·2-s − 0.0757·4-s − 0.514·5-s − 0.665·7-s − 1.03·8-s − 0.495·10-s + 0.301·11-s − 0.564·13-s − 0.640·14-s − 0.918·16-s + 1.10·17-s + 1.61·19-s + 0.0389·20-s + 0.289·22-s − 0.557·23-s − 0.734·25-s − 0.542·26-s + 0.0504·28-s − 1.12·29-s + 1.52·31-s + 0.151·32-s + 1.06·34-s + 0.342·35-s + 0.729·37-s + 1.55·38-s + 0.532·40-s + 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2673 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2673 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.945174725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.945174725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 4.55T + 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 6.74T + 73T^{2} \) |
| 79 | \( 1 - 1.60T + 79T^{2} \) |
| 83 | \( 1 - 4.98T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
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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
−8.921703201121716861694110429507, −7.891481441133043835549464761017, −7.35727576109588449818506036145, −6.29255569070666713633195501248, −5.65769712421164646794885810895, −4.92647019239185730260439910737, −3.93391668220929779614302378018, −3.45220897119758923309240678282, −2.53312788957420812509891886907, −0.74833888982828642016509566297,
0.74833888982828642016509566297, 2.53312788957420812509891886907, 3.45220897119758923309240678282, 3.93391668220929779614302378018, 4.92647019239185730260439910737, 5.65769712421164646794885810895, 6.29255569070666713633195501248, 7.35727576109588449818506036145, 7.891481441133043835549464761017, 8.921703201121716861694110429507
