| L(s) = 1 | + (1.13 + 0.840i)2-s + (0.198 − 0.0223i)3-s + (0.587 + 1.91i)4-s + (1.17 − 2.43i)5-s + (0.244 + 0.141i)6-s + (−0.635 − 0.506i)7-s + (−0.938 + 2.66i)8-s + (−2.88 + 0.658i)9-s + (3.37 − 1.78i)10-s + (−3.17 + 1.99i)11-s + (0.159 + 0.366i)12-s + (1.70 + 0.389i)13-s + (−0.296 − 1.10i)14-s + (0.178 − 0.509i)15-s + (−3.31 + 2.24i)16-s + (0.662 − 0.662i)17-s + ⋯ |
| L(s) = 1 | + (0.804 + 0.594i)2-s + (0.114 − 0.0129i)3-s + (0.293 + 0.955i)4-s + (0.524 − 1.08i)5-s + (0.0999 + 0.0577i)6-s + (−0.240 − 0.191i)7-s + (−0.331 + 0.943i)8-s + (−0.961 + 0.219i)9-s + (1.06 − 0.563i)10-s + (−0.958 + 0.601i)11-s + (0.0460 + 0.105i)12-s + (0.473 + 0.108i)13-s + (−0.0792 − 0.296i)14-s + (0.0460 − 0.131i)15-s + (−0.827 + 0.561i)16-s + (0.160 − 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48748 + 0.475323i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48748 + 0.475323i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.13 - 0.840i)T \) |
| 29 | \( 1 + (-3.86 - 3.74i)T \) |
| good | 3 | \( 1 + (-0.198 + 0.0223i)T + (2.92 - 0.667i)T^{2} \) |
| 5 | \( 1 + (-1.17 + 2.43i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (0.635 + 0.506i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (3.17 - 1.99i)T + (4.77 - 9.91i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 0.389i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.662 + 0.662i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.725 + 6.43i)T + (-18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (2.06 + 4.28i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-2.22 - 6.34i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (1.57 - 2.50i)T + (-16.0 - 33.3i)T^{2} \) |
| 41 | \( 1 + (3.80 + 3.80i)T + 41iT^{2} \) |
| 43 | \( 1 + (-11.2 - 3.93i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-0.750 - 1.19i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (2.81 + 1.35i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + 6.23iT - 59T^{2} \) |
| 61 | \( 1 + (-1.69 - 15.0i)T + (-59.4 + 13.5i)T^{2} \) |
| 67 | \( 1 + (-2.86 - 12.5i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (-1.76 + 7.71i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (3.40 + 1.19i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (-3.21 + 5.11i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (5.17 - 4.12i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.84 + 0.994i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (-0.706 + 6.27i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−13.57340493476070244504120019499, −12.92527234842979927065010337373, −11.98269388504159445072853921642, −10.64536665487964783186088537510, −9.032588744317934677100483239883, −8.265274603452086490546962798041, −6.85872666992541752009079473404, −5.48572520753846605724623220796, −4.68954987082961923541141496729, −2.74831469778543427317458689825,
2.52477366055217819220814694731, 3.55609219811054088404031076641, 5.69172167143679072622825075512, 6.19782181087755578621244041517, 7.972110244254159011756885509691, 9.620032079417284873028790421363, 10.53561672864141120329417249607, 11.33881362141701996655309808697, 12.43642778933880961894866673828, 13.72999292864596703425737143242
