L(s) = 1 | + (0.450 − 1.38i)2-s + (−0.910 − 0.661i)4-s + (−0.574 − 1.76i)5-s + (−0.148 + 0.107i)8-s + (0.309 − 0.951i)9-s − 2.71·10-s + (0.728 + 0.684i)11-s + (−0.613 + 1.88i)13-s + (−0.265 − 0.816i)16-s + (−1.17 − 0.856i)18-s + (−0.866 + 0.629i)19-s + (−0.646 + 1.99i)20-s + (1.27 − 0.702i)22-s + 1.07·23-s + (−1.98 + 1.44i)25-s + (2.34 + 1.70i)26-s + ⋯ |
L(s) = 1 | + (0.450 − 1.38i)2-s + (−0.910 − 0.661i)4-s + (−0.574 − 1.76i)5-s + (−0.148 + 0.107i)8-s + (0.309 − 0.951i)9-s − 2.71·10-s + (0.728 + 0.684i)11-s + (−0.613 + 1.88i)13-s + (−0.265 − 0.816i)16-s + (−1.17 − 0.856i)18-s + (−0.866 + 0.629i)19-s + (−0.646 + 1.99i)20-s + (1.27 − 0.702i)22-s + 1.07·23-s + (−1.98 + 1.44i)25-s + (2.34 + 1.70i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.151952758\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.151952758\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.728 - 0.684i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - 1.07T + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - 1.93T + T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + 0.374T + T^{2} \) |
| 97 | \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \) |
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
−9.728618885739768697760874218946, −9.420653516879937837791747119625, −8.742078511208344147601464570465, −7.46950130392146725999804423987, −6.45555956879143442403078090552, −4.94347646534228555793947934513, −4.21268666013945840663620444448, −3.93172242933181099076793543865, −2.08493534200915078985676191043, −1.12987056450597520703206116259,
2.63346744609971456760910063111, 3.56078025551242301361432562019, 4.79976916359181485400511344396, 5.67481424378641626843375832750, 6.73728004498547108478824921843, 7.06515069330101932701598102572, 7.943836647320558442959048355531, 8.526118408321618985353522655045, 10.15081031400687888130434791883, 10.75822767066634528324959999864