| L(s) = 1 | + (0.540 + 0.841i)2-s + (2.80 + 0.402i)3-s + (−0.415 + 0.909i)4-s + (0.225 − 2.22i)5-s + (1.17 + 2.57i)6-s + (−2.07 + 1.79i)7-s + (−0.989 + 0.142i)8-s + (4.81 + 1.41i)9-s + (1.99 − 1.01i)10-s + (−2.81 − 1.80i)11-s + (−1.53 + 2.38i)12-s + (2.48 + 2.15i)13-s + (−2.63 − 0.774i)14-s + (1.52 − 6.14i)15-s + (−0.654 − 0.755i)16-s + (−3.09 + 1.41i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (1.61 + 0.232i)3-s + (−0.207 + 0.454i)4-s + (0.100 − 0.994i)5-s + (0.480 + 1.05i)6-s + (−0.785 + 0.680i)7-s + (−0.349 + 0.0503i)8-s + (1.60 + 0.471i)9-s + (0.630 − 0.320i)10-s + (−0.848 − 0.545i)11-s + (−0.441 + 0.687i)12-s + (0.689 + 0.597i)13-s + (−0.704 − 0.206i)14-s + (0.394 − 1.58i)15-s + (−0.163 − 0.188i)16-s + (−0.751 + 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96655 + 0.811402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96655 + 0.811402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 5 | \( 1 + (-0.225 + 2.22i)T \) |
| 23 | \( 1 + (4.50 - 1.65i)T \) |
| good | 3 | \( 1 + (-2.80 - 0.402i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (2.07 - 1.79i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (2.81 + 1.80i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.48 - 2.15i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.09 - 1.41i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.40 + 7.46i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.0669 + 0.146i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.0425 + 0.296i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.77 - 9.45i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.66 + 1.07i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-5.68 - 0.816i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 11.2iT - 47T^{2} \) |
| 53 | \( 1 + (3.34 - 2.89i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.88 + 3.32i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.444 - 3.09i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.45 - 5.37i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (7.55 - 4.85i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.43 - 1.56i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.12 + 4.75i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.60 + 15.6i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.04 - 7.27i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.893 - 3.04i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−12.88752543597045902591062569110, −11.60051583977001452919580141866, −9.900708383362557328784254168186, −8.890987364641272040104919210026, −8.705542142579982026893276270554, −7.56818642683182333676836004849, −6.17691418146903261525074371837, −4.87605845911497838736764974900, −3.64907210201060507524509823967, −2.46846272613719290998357470135,
2.15353692045450461323992492623, 3.21089849933259383941614854027, 3.95174469535607645081342803349, 6.00780318256760351712289374412, 7.30526537298963673548095388714, 8.019475336177098501339203220741, 9.432787822483470014870144489323, 10.12162975953990897401084779617, 10.86576284539509959236330847502, 12.46792375870691365693370311397
