L(s) = 1 | + (0.0911 − 0.663i)2-s + (0.519 + 0.854i)3-s + (0.530 + 0.148i)4-s + (0.614 − 0.266i)6-s + (0.413 − 0.953i)8-s + (−0.460 + 0.887i)9-s + (0.148 + 0.530i)12-s + (−0.123 − 0.0750i)16-s + (−0.0997 + 0.0931i)17-s + (0.547 + 0.386i)18-s + (0.644 − 1.81i)19-s + (0.269 + 1.96i)23-s + (1.02 − 0.141i)24-s + (−0.997 + 0.0682i)27-s + (1.16 + 0.709i)31-s + (0.594 − 0.730i)32-s + ⋯ |
L(s) = 1 | + (0.0911 − 0.663i)2-s + (0.519 + 0.854i)3-s + (0.530 + 0.148i)4-s + (0.614 − 0.266i)6-s + (0.413 − 0.953i)8-s + (−0.460 + 0.887i)9-s + (0.148 + 0.530i)12-s + (−0.123 − 0.0750i)16-s + (−0.0997 + 0.0931i)17-s + (0.547 + 0.386i)18-s + (0.644 − 1.81i)19-s + (0.269 + 1.96i)23-s + (1.02 − 0.141i)24-s + (−0.997 + 0.0682i)27-s + (1.16 + 0.709i)31-s + (0.594 − 0.730i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.973337895\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.973337895\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.519 - 0.854i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (0.979 + 0.203i)T \) |
good | 2 | \( 1 + (-0.0911 + 0.663i)T + (-0.962 - 0.269i)T^{2} \) |
| 7 | \( 1 + (-0.990 - 0.136i)T^{2} \) |
| 11 | \( 1 + (0.0682 + 0.997i)T^{2} \) |
| 13 | \( 1 + (-0.334 + 0.942i)T^{2} \) |
| 17 | \( 1 + (0.0997 - 0.0931i)T + (0.0682 - 0.997i)T^{2} \) |
| 19 | \( 1 + (-0.644 + 1.81i)T + (-0.775 - 0.631i)T^{2} \) |
| 23 | \( 1 + (-0.269 - 1.96i)T + (-0.962 + 0.269i)T^{2} \) |
| 29 | \( 1 + (0.334 + 0.942i)T^{2} \) |
| 31 | \( 1 + (-1.16 - 0.709i)T + (0.460 + 0.887i)T^{2} \) |
| 37 | \( 1 + (-0.917 + 0.398i)T^{2} \) |
| 41 | \( 1 + (-0.682 + 0.730i)T^{2} \) |
| 43 | \( 1 + (0.854 + 0.519i)T^{2} \) |
| 53 | \( 1 + (0.366 + 0.843i)T + (-0.682 + 0.730i)T^{2} \) |
| 59 | \( 1 + (-0.854 + 0.519i)T^{2} \) |
| 61 | \( 1 + (0.234 - 1.12i)T + (-0.917 - 0.398i)T^{2} \) |
| 67 | \( 1 + (-0.990 + 0.136i)T^{2} \) |
| 71 | \( 1 + (-0.962 + 0.269i)T^{2} \) |
| 73 | \( 1 + (-0.576 + 0.816i)T^{2} \) |
| 79 | \( 1 + (-1.05 - 0.861i)T + (0.203 + 0.979i)T^{2} \) |
| 83 | \( 1 + (1.34 + 1.25i)T + (0.0682 + 0.997i)T^{2} \) |
| 89 | \( 1 + (0.775 - 0.631i)T^{2} \) |
| 97 | \( 1 + (0.460 - 0.887i)T^{2} \) |
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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
−9.000856828874549693533304872659, −8.077556204856023265112326735352, −7.32926301117318319792592394816, −6.68419077831450072649447077922, −5.51651394830233890009215577309, −4.79586247347863507067931026513, −3.90662887805695802367822059411, −3.11870238792808194162843925110, −2.58511966233142358569870445246, −1.40286385826331552082027944050,
1.23233745187348154411750817570, 2.24870290978534386007704341481, 2.98878925642497746442890875547, 4.12139150897818731887948351357, 5.19995395885202073820859759127, 6.11790925490211979332183872270, 6.46164242319114794756225347451, 7.25239239267639351301922493706, 8.069736792055101540974336210343, 8.230427914113594721962774999461