| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (−1.71 + 0.372i)3-s + (−0.989 − 0.142i)4-s + (−1.89 − 1.19i)5-s + (−0.249 − 1.73i)6-s + (3.16 − 1.73i)7-s + (0.212 − 0.977i)8-s + (0.0607 − 0.0277i)9-s + (1.32 − 1.80i)10-s + (−1.69 − 1.47i)11-s + (1.74 − 0.124i)12-s + (2.52 − 4.63i)13-s + (1.50 + 3.28i)14-s + (3.68 + 1.33i)15-s + (0.959 + 0.281i)16-s + (2.07 + 1.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (−0.987 + 0.214i)3-s + (−0.494 − 0.0711i)4-s + (−0.846 − 0.532i)5-s + (−0.101 − 0.707i)6-s + (1.19 − 0.654i)7-s + (0.0751 − 0.345i)8-s + (0.0202 − 0.00924i)9-s + (0.418 − 0.569i)10-s + (−0.511 − 0.443i)11-s + (0.504 − 0.0360i)12-s + (0.701 − 1.28i)13-s + (0.400 + 0.877i)14-s + (0.950 + 0.344i)15-s + (0.239 + 0.0704i)16-s + (0.502 + 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.572388 - 0.245750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.572388 - 0.245750i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 5 | \( 1 + (1.89 + 1.19i)T \) |
| 23 | \( 1 + (2.42 + 4.14i)T \) |
| good | 3 | \( 1 + (1.71 - 0.372i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-3.16 + 1.73i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (1.69 + 1.47i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.52 + 4.63i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-2.07 - 1.55i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.731 + 5.08i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (4.62 - 0.664i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (7.65 - 4.92i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.86 + 1.06i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-1.83 + 4.02i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (2.56 + 11.7i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (0.0966 + 0.0966i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.05 - 7.42i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.72 + 5.87i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.06 - 9.43i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-9.75 - 0.697i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (3.94 + 4.55i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-4.01 - 5.36i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-2.52 + 0.742i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.61 - 4.32i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-11.7 - 7.58i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.93 - 7.86i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.01090750190686832108731692081, −10.78709118160911187156484828849, −10.73951319545418901660138067809, −8.731522657865190970651311647709, −8.080514323492538884399277387817, −7.16726517435360515929564259949, −5.55138557863508944058229901386, −5.09283065458130075507993423228, −3.81605895599579870162173472316, −0.61756374021485597480520758687,
1.81599112121407061806726802777, 3.68514218631602079879341044847, 4.96774209535870647171008037409, 6.00919617507277953956968140637, 7.46402328898575946770983458154, 8.330498899362126049749219476783, 9.615111308367212068922396861482, 10.90695566705301319740339181224, 11.61792241023715049085273637606, 11.72992121784767752719310367103
