| L(s) = 1 | + (0.998 + 0.0523i)2-s + (−1.11 − 1.38i)3-s + (0.994 + 0.104i)4-s + (−2.04 − 0.906i)5-s + (−1.04 − 1.43i)6-s + (−2.64 + 0.155i)7-s + (0.987 + 0.156i)8-s + (−0.0346 + 0.163i)9-s + (−1.99 − 1.01i)10-s + (−1.16 + 0.247i)11-s + (−0.969 − 1.49i)12-s + (−4.00 − 2.04i)13-s + (−2.64 + 0.0166i)14-s + (1.03 + 3.84i)15-s + (0.978 + 0.207i)16-s + (0.426 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.0370i)2-s + (−0.646 − 0.798i)3-s + (0.497 + 0.0522i)4-s + (−0.914 − 0.405i)5-s + (−0.427 − 0.587i)6-s + (−0.998 + 0.0586i)7-s + (0.349 + 0.0553i)8-s + (−0.0115 + 0.0543i)9-s + (−0.630 − 0.320i)10-s + (−0.351 + 0.0746i)11-s + (−0.279 − 0.430i)12-s + (−1.11 − 0.566i)13-s + (−0.707 + 0.00445i)14-s + (0.267 + 0.992i)15-s + (0.244 + 0.0519i)16-s + (0.103 + 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.167476 - 0.735688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.167476 - 0.735688i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 - 0.0523i)T \) |
| 5 | \( 1 + (2.04 + 0.906i)T \) |
| 7 | \( 1 + (2.64 - 0.155i)T \) |
| good | 3 | \( 1 + (1.11 + 1.38i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (1.16 - 0.247i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (4.00 + 2.04i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.426 - 1.11i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.156 + 1.49i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.341 + 6.52i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (2.18 - 3.00i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.20 + 4.95i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.14 + 2.69i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-8.58 + 2.79i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (0.920 + 0.920i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.57 + 1.75i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-2.45 + 1.98i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (7.47 - 8.30i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-11.1 + 10.0i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (8.98 - 3.45i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-1.52 - 1.10i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.64 + 7.15i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (4.88 + 10.9i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (1.36 - 8.62i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-4.54 - 5.05i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.495 - 3.12i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.44262418677420054225716958621, −10.43668876618974332444499528467, −9.243452920082276639940941072911, −7.87091390845329921372868078490, −7.12487886604439900690787240383, −6.25213278597881773604965482170, −5.20882003402326644057978596203, −4.01955659900607989507901706340, −2.68419057304601516600910073863, −0.42707875682831548127134846405,
2.81108411260601361971652000976, 3.93017963331846343718282704438, 4.81219694042907144612949604766, 5.88324725335927047867374708434, 7.01029939921488479592951541588, 7.82369341073040699923227172170, 9.521700825710572309563747308748, 10.16465206458235216277232488259, 11.15204853254243791365446937668, 11.76300259276209232479500507130
