| L(s) = 1 | + (−0.984 − 0.173i)2-s + (0.826 − 1.52i)3-s + (0.939 + 0.342i)4-s + (−1.99 + 1.67i)5-s + (−1.07 + 1.35i)6-s + (1.84 + 1.90i)7-s + (−0.866 − 0.5i)8-s + (−1.63 − 2.51i)9-s + (2.25 − 1.30i)10-s + (1.10 − 1.31i)11-s + (1.29 − 1.14i)12-s + (6.54 − 1.15i)13-s + (−1.48 − 2.19i)14-s + (0.902 + 4.42i)15-s + (0.766 + 0.642i)16-s + (3.19 + 5.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (0.476 − 0.878i)3-s + (0.469 + 0.171i)4-s + (−0.893 + 0.749i)5-s + (−0.440 + 0.553i)6-s + (0.695 + 0.718i)7-s + (−0.306 − 0.176i)8-s + (−0.545 − 0.838i)9-s + (0.714 − 0.412i)10-s + (0.331 − 0.395i)11-s + (0.374 − 0.331i)12-s + (1.81 − 0.320i)13-s + (−0.396 − 0.585i)14-s + (0.232 + 1.14i)15-s + (0.191 + 0.160i)16-s + (0.774 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11882 - 0.245161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11882 - 0.245161i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (-0.826 + 1.52i)T \) |
| 7 | \( 1 + (-1.84 - 1.90i)T \) |
| good | 5 | \( 1 + (1.99 - 1.67i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-1.10 + 1.31i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-6.54 + 1.15i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.19 - 5.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.36 + 0.789i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.89 + 7.94i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.93 - 0.693i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.301 + 0.829i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (2.57 + 4.45i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.142 + 0.807i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.82 - 5.72i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (6.60 - 2.40i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 6.75iT - 53T^{2} \) |
| 59 | \( 1 + (-0.690 + 0.579i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.46 + 4.02i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.06 + 11.7i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.69 + 1.55i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.04 - 0.605i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.88 - 10.6i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.170 + 0.965i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (1.37 - 2.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.7 - 14.0i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.00689724708743396634396388840, −10.85697007114370757427892572414, −9.068135490112175652927401315109, −8.277338415760990155049360563655, −7.992530410597709862944359801976, −6.65626205476732589205477181836, −5.99012884416456813259499215257, −3.83376303716749757411822998726, −2.81904762145785731201465829503, −1.28747263950361366049173753043,
1.27051580167114729919961729352, 3.43812810320731517685316893537, 4.34200357720016108886151671139, 5.39524566105106645289666352019, 7.07724839214094221330687320407, 8.024864862793265542383184516826, 8.609166371555573913198266128285, 9.454899350703763997266369423521, 10.42183525595247580882098770587, 11.38112480647553900908796556314
