| L(s) = 1 | + (1.00 − 0.217i)3-s + (2.23 − 0.0439i)5-s + (2.59 − 1.41i)7-s + (−1.77 + 0.810i)9-s + (−0.536 − 0.465i)11-s + (2.46 − 4.52i)13-s + (2.22 − 0.530i)15-s + (1.03 + 0.773i)17-s + (−0.316 + 2.19i)19-s + (2.28 − 1.98i)21-s + (−0.539 − 4.76i)23-s + (4.99 − 0.196i)25-s + (−4.05 + 3.03i)27-s + (−0.503 + 0.0724i)29-s + (−1.36 + 0.877i)31-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.125i)3-s + (0.999 − 0.0196i)5-s + (0.980 − 0.535i)7-s + (−0.591 + 0.270i)9-s + (−0.161 − 0.140i)11-s + (0.684 − 1.25i)13-s + (0.575 − 0.137i)15-s + (0.250 + 0.187i)17-s + (−0.0725 + 0.504i)19-s + (0.499 − 0.432i)21-s + (−0.112 − 0.993i)23-s + (0.999 − 0.0393i)25-s + (−0.781 + 0.584i)27-s + (−0.0935 + 0.0134i)29-s + (−0.245 + 0.157i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.34845 - 0.603104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34845 - 0.603104i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.0439i)T \) |
| 23 | \( 1 + (0.539 + 4.76i)T \) |
| good | 3 | \( 1 + (-1.00 + 0.217i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-2.59 + 1.41i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (0.536 + 0.465i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.46 + 4.52i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 0.773i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.316 - 2.19i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.503 - 0.0724i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.36 - 0.877i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.90 - 1.82i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (5.21 - 11.4i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.11 + 5.14i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (2.49 + 2.49i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.01 - 3.68i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (0.562 + 1.91i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.265 - 0.413i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.201 - 0.0143i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (0.714 + 0.824i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.54 - 4.73i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-12.2 + 3.59i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.98 + 5.32i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.52 + 0.982i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.46 - 14.6i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.16964436573236957920423374325, −9.048119132619152291012675120255, −8.175156003753204765421614061005, −7.88766919802330686590685805834, −6.49402954078185928739275668441, −5.63111505887232416156437038378, −4.84725026387208982910811798738, −3.45620311671795963468518593893, −2.43330017408867159641821931938, −1.25935950538153254445350959791,
1.65079413209672110380289225629, 2.48432852518814813963179860856, 3.74405049305672974590814268125, 4.98099794436422165178132535916, 5.74697597430916561399680315778, 6.65212633901555150918462092912, 7.78298980856258731074096937041, 8.737504996635001939427742326296, 9.149343021924475893399741250159, 9.916342705378082780271961500723
