| L(s) = 1 | + (1.41 + 0.0989i)2-s + (−0.214 − 2.18i)3-s + (1.98 + 0.279i)4-s + (−0.223 + 0.418i)5-s + (−0.0872 − 3.09i)6-s + (1.27 + 2.31i)7-s + (2.76 + 0.589i)8-s + (−1.76 + 0.351i)9-s + (−0.357 + 0.568i)10-s + (0.770 − 0.632i)11-s + (0.183 − 4.37i)12-s + (1.79 − 0.959i)13-s + (1.56 + 3.39i)14-s + (0.961 + 0.398i)15-s + (3.84 + 1.10i)16-s + (−0.649 + 0.268i)17-s + ⋯ |
| L(s) = 1 | + (0.997 + 0.0699i)2-s + (−0.124 − 1.25i)3-s + (0.990 + 0.139i)4-s + (−0.100 + 0.187i)5-s + (−0.0356 − 1.26i)6-s + (0.481 + 0.876i)7-s + (0.978 + 0.208i)8-s + (−0.589 + 0.117i)9-s + (−0.113 + 0.179i)10-s + (0.232 − 0.190i)11-s + (0.0530 − 1.26i)12-s + (0.497 − 0.266i)13-s + (0.419 + 0.907i)14-s + (0.248 + 0.102i)15-s + (0.961 + 0.276i)16-s + (−0.157 + 0.0652i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.93188 - 1.11054i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.93188 - 1.11054i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0989i)T \) |
| 7 | \( 1 + (-1.27 - 2.31i)T \) |
| good | 3 | \( 1 + (0.214 + 2.18i)T + (-2.94 + 0.585i)T^{2} \) |
| 5 | \( 1 + (0.223 - 0.418i)T + (-2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (-0.770 + 0.632i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.79 + 0.959i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (0.649 - 0.268i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.434 + 1.43i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.89 + 2.83i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (5.31 + 4.36i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-4.48 - 4.48i)T + 31iT^{2} \) |
| 37 | \( 1 + (7.92 - 2.40i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (4.22 + 2.82i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (10.5 + 1.03i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-8.64 + 3.58i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.51 + 2.06i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (2.86 - 5.36i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (0.413 + 4.19i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.0725 - 0.736i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (1.21 - 6.13i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (6.08 - 1.20i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.88 + 6.97i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-11.6 - 3.52i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (6.00 + 8.99i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (12.5 - 12.5i)T - 97iT^{2} \) |
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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
−10.35787204725164122616000915103, −8.834390691078666347348061532846, −8.137103706915803496416627025366, −7.15802878276330394064099276837, −6.59012766273494165226412541366, −5.71789857745951756581581074003, −4.90507066773151495897096426247, −3.52085773313701805762953282072, −2.42260700268416481182117152865, −1.41965426776164961420958067576,
1.56801199359630205484749700472, 3.30831408950153684313809576882, 4.05407155423106760696437075107, 4.70714993599383924413430407202, 5.46945904637308449879591129197, 6.64982346132988842107590291424, 7.48469196731117788341910738209, 8.594047462864051228421345901457, 9.680238385761566689542037377653, 10.42330643718978581418681477550
