L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−2 + i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s − 3i·9-s + (2.12 + 0.707i)10-s + (−3 + 1.41i)11-s + (1.41 − 1.41i)13-s − 1.00i·14-s − 1.00·16-s + (−2.12 + 2.12i)18-s + 4.24·19-s + (−1.00 − 2.00i)20-s + (3.12 + 1.12i)22-s + (−2 − 2i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.894 + 0.447i)5-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − i·9-s + (0.670 + 0.223i)10-s + (−0.904 + 0.426i)11-s + (0.392 − 0.392i)13-s − 0.267i·14-s − 0.250·16-s + (−0.499 + 0.499i)18-s + 0.973·19-s + (−0.223 − 0.447i)20-s + (0.665 + 0.239i)22-s + (−0.417 − 0.417i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.731652 - 0.511481i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.731652 - 0.511481i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (3 - 1.41i)T \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 13 | \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + (2 + 2i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-6 + 6i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (-5.65 + 5.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7 + 7i)T - 47iT^{2} \) |
| 53 | \( 1 + (4 + 4i)T + 53iT^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 2.82iT - 61T^{2} \) |
| 67 | \( 1 + (3 - 3i)T - 67iT^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-4.24 + 4.24i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 + (-6 + 6i)T - 97iT^{2} \) |
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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
−10.31005969221271948082869424983, −9.325559132272720668137035667683, −8.442971184543783040033195167214, −7.71112599086515889039053184778, −6.93426699676010219117513401201, −5.75330273446599044718304449550, −4.44463948372988282171460184951, −3.44797687424685792106139419462, −2.50947526449916942830668017152, −0.65530401066015364356566676435,
1.13953218114641707511722086791, 2.86556166036080567521721632173, 4.35821044721616690263370439198, 5.08163076776503735119210157954, 6.11739477291568467300617191012, 7.43897779080180237454815393021, 7.86978718722745349747725999861, 8.508103623705158429010723981047, 9.527878312529593958592503526275, 10.52725588954706209720894518679