| L(s) = 1 | + (0.450 − 0.0980i)3-s + (1.16 + 1.90i)5-s + (−3.36 + 1.83i)7-s + (−2.53 + 1.15i)9-s + (−0.179 − 0.155i)11-s + (−2.67 + 4.88i)13-s + (0.713 + 0.744i)15-s + (−1.75 − 1.31i)17-s + (0.184 − 1.28i)19-s + (−1.33 + 1.15i)21-s + (3.96 − 2.69i)23-s + (−2.26 + 4.45i)25-s + (−2.13 + 1.59i)27-s + (0.986 − 0.141i)29-s + (7.10 − 4.56i)31-s + ⋯ |
| L(s) = 1 | + (0.260 − 0.0565i)3-s + (0.522 + 0.852i)5-s + (−1.27 + 0.694i)7-s + (−0.845 + 0.385i)9-s + (−0.0540 − 0.0467i)11-s + (−0.740 + 1.35i)13-s + (0.184 + 0.192i)15-s + (−0.425 − 0.318i)17-s + (0.0422 − 0.293i)19-s + (−0.291 + 0.252i)21-s + (0.827 − 0.561i)23-s + (−0.453 + 0.891i)25-s + (−0.411 + 0.307i)27-s + (0.183 − 0.0263i)29-s + (1.27 − 0.819i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.558752 + 0.880034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.558752 + 0.880034i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.16 - 1.90i)T \) |
| 23 | \( 1 + (-3.96 + 2.69i)T \) |
| good | 3 | \( 1 + (-0.450 + 0.0980i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (3.36 - 1.83i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (0.179 + 0.155i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.67 - 4.88i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (1.75 + 1.31i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.184 + 1.28i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.986 + 0.141i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.10 + 4.56i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-5.10 - 1.90i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.72 - 3.76i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.31 - 6.06i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-2.92 - 2.92i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.65 - 10.3i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-2.27 - 7.73i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.294 + 0.458i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (6.12 + 0.437i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (7.22 + 8.33i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (5.79 + 7.73i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-8.54 + 2.50i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.344 + 0.922i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-10.6 - 6.81i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.85 - 4.97i)T + (-73.3 + 63.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.38816722189184204387092134318, −10.36274446965021929386422486704, −9.386206696143204327661746783570, −8.993887943783530582638040378684, −7.56946221695036610569601022166, −6.54451445948973749405223611311, −6.01742395653555489413589922523, −4.61199056715663914711567066388, −2.91113188675954591846872384206, −2.48052310664633713365049313820,
0.59136899294857678713284035051, 2.67769257939019735344299965533, 3.70507264265946157445112843806, 5.12533409444480746741374994945, 5.99741893750288900699381143559, 7.03306666021447707575294060935, 8.214687798913891128783571316838, 9.020808192714495898468496987317, 9.909266671982779916884565868225, 10.43102186533536041868415410724
