| L(s) = 1 | − 2·4-s + 4.94·7-s − 6.47·13-s + 4·16-s − 5·25-s − 9.88·28-s − 9.74·31-s − 3.20·37-s − 0.551·43-s + 17.4·49-s + 12.9·52-s − 4.39·61-s − 8·64-s + 16.2·67-s − 15.3·73-s + 9.68·79-s − 32.0·91-s − 19·97-s + 10·100-s + 0.0614·103-s − 19·109-s + 19.7·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.86·7-s − 1.79·13-s + 16-s − 25-s − 1.86·28-s − 1.75·31-s − 0.527·37-s − 0.0841·43-s + 2.49·49-s + 1.79·52-s − 0.562·61-s − 64-s + 1.98·67-s − 1.80·73-s + 1.08·79-s − 3.35·91-s − 1.92·97-s + 100-s + 0.00605·103-s − 1.81·109-s + 1.86·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 9.74T + 31T^{2} \) |
| 37 | \( 1 + 3.20T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 0.551T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 19T + 97T^{2} \) |
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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
−8.162946445059610228617676525780, −7.73456126014509107803101885344, −7.05816245167145613336613530527, −5.57056943180987743156010355710, −5.18192511537047999527348466971, −4.51820719072461772783824911208, −3.78562209295169922754630577625, −2.38470356321776529262291628395, −1.49498294532556282548389047200, 0,
1.49498294532556282548389047200, 2.38470356321776529262291628395, 3.78562209295169922754630577625, 4.51820719072461772783824911208, 5.18192511537047999527348466971, 5.57056943180987743156010355710, 7.05816245167145613336613530527, 7.73456126014509107803101885344, 8.162946445059610228617676525780
