| L(s) = 1 | + (0.212 − 0.977i)2-s + (−1.00 + 0.755i)3-s + (−0.909 − 0.415i)4-s + (−2.23 − 0.105i)5-s + (0.523 + 1.14i)6-s + (−0.128 + 1.79i)7-s + (−0.599 + 0.800i)8-s + (−0.397 + 1.35i)9-s + (−0.578 + 2.16i)10-s + (−0.836 + 1.30i)11-s + (1.23 − 0.267i)12-s + (−1.97 + 0.141i)13-s + (1.72 + 0.507i)14-s + (2.33 − 1.58i)15-s + (0.654 + 0.755i)16-s + (−2.48 + 6.65i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 0.690i)2-s + (−0.582 + 0.436i)3-s + (−0.454 − 0.207i)4-s + (−0.998 − 0.0473i)5-s + (0.213 + 0.467i)6-s + (−0.0485 + 0.678i)7-s + (−0.211 + 0.283i)8-s + (−0.132 + 0.451i)9-s + (−0.182 + 0.683i)10-s + (−0.252 + 0.392i)11-s + (0.355 − 0.0773i)12-s + (−0.547 + 0.0391i)13-s + (0.461 + 0.135i)14-s + (0.602 − 0.407i)15-s + (0.163 + 0.188i)16-s + (−0.602 + 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.268138 + 0.345120i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.268138 + 0.345120i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.212 + 0.977i)T \) |
| 5 | \( 1 + (2.23 + 0.105i)T \) |
| 23 | \( 1 + (4.36 + 1.99i)T \) |
| good | 3 | \( 1 + (1.00 - 0.755i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (0.128 - 1.79i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (0.836 - 1.30i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.97 - 0.141i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (2.48 - 6.65i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.28 + 2.81i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.50 - 0.685i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.456 + 3.17i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.675 - 1.23i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-6.67 + 1.95i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.656 - 0.877i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (8.44 - 8.44i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.60 - 0.186i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (4.49 + 3.89i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (2.10 - 0.302i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-9.26 - 2.01i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-5.43 + 3.49i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (13.7 - 5.12i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (2.07 - 2.39i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (10.4 - 5.73i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (2.68 - 18.6i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-9.83 - 5.36i)T + (52.4 + 81.6i)T^{2} \) |
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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
−12.35551609432742173015629922281, −11.42338687273540556244371659152, −10.83387676244644756147682970622, −9.844451424588484656746666542280, −8.642279647712252613485886755462, −7.70773864958986039857447385582, −6.09867534087160860766929639356, −4.90403485115256398769731330267, −4.04808164809197652160906686372, −2.40148265941137347605744466784,
0.35741362753006433679789395294, 3.37741116966099527260213452715, 4.62346727398821220504752063692, 5.84702281499989955764544314080, 7.07681603923284534274441519518, 7.52444599312467738533507655568, 8.757621044842838012352802790590, 9.964795252854092055893260980261, 11.30685132166297880374437021587, 11.89635350276741561542629048386
