L(s) = 1 | + (0.190 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.5 + 0.363i)4-s + (−0.5 + 0.363i)6-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.190 − 0.587i)12-s + (1.30 − 0.951i)17-s + 0.618·18-s + (−0.5 + 0.363i)19-s + (−0.5 + 1.53i)23-s − 24-s + (0.309 − 0.951i)27-s + (1.30 − 0.951i)31-s + 32-s + ⋯ |
L(s) = 1 | + (0.190 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.5 + 0.363i)4-s + (−0.5 + 0.363i)6-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.190 − 0.587i)12-s + (1.30 − 0.951i)17-s + 0.618·18-s + (−0.5 + 0.363i)19-s + (−0.5 + 1.53i)23-s − 24-s + (0.309 − 0.951i)27-s + (1.30 − 0.951i)31-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.223542271\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.223542271\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.678289676755534710263529480070, −8.183924939145434549514618016497, −7.65286741830192770203839697203, −6.97623119272940393536961514362, −6.09311679096733660154682965030, −5.33134614210153815006745302265, −4.28719062196416469105712941800, −3.27590939591069862672723036200, −2.23265395492047832501830480390, −1.19308079420884992126942837988,
1.27712741661469180757928369849, 2.75476685527093939894476760669, 4.06275255300185817716804635534, 4.82717095527255297819834355556, 5.65129978063291165485918263270, 6.30417141098339279055717181142, 6.84709292687354781598508739092, 7.906816252585620467824948760172, 8.642123415979426067648382299433, 9.790662148730172407653017096026