L(s) = 1 | + (−0.521 + 0.853i)3-s + (0.170 − 0.985i)5-s + (0.793 + 0.607i)7-s + (−0.455 − 0.890i)9-s + (0.348 + 0.937i)11-s + (0.662 − 0.748i)13-s + (0.751 + 0.659i)15-s + (−0.994 + 0.108i)17-s + (0.828 − 0.560i)19-s + (−0.932 + 0.360i)21-s + (−0.970 + 0.240i)23-s + (−0.941 − 0.336i)25-s + (0.997 + 0.0753i)27-s + (−0.630 − 0.775i)29-s + (−0.154 + 0.988i)31-s + ⋯ |
L(s) = 1 | + (−0.521 + 0.853i)3-s + (0.170 − 0.985i)5-s + (0.793 + 0.607i)7-s + (−0.455 − 0.890i)9-s + (0.348 + 0.937i)11-s + (0.662 − 0.748i)13-s + (0.751 + 0.659i)15-s + (−0.994 + 0.108i)17-s + (0.828 − 0.560i)19-s + (−0.932 + 0.360i)21-s + (−0.970 + 0.240i)23-s + (−0.941 − 0.336i)25-s + (0.997 + 0.0753i)27-s + (−0.630 − 0.775i)29-s + (−0.154 + 0.988i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02807565293 + 0.4348671055i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02807565293 + 0.4348671055i\) |
\(L(1)\) |
\(\approx\) |
\(0.8384409846 + 0.1733934409i\) |
\(L(1)\) |
\(\approx\) |
\(0.8384409846 + 0.1733934409i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 3 | \( 1 + (-0.521 + 0.853i)T \) |
| 5 | \( 1 + (0.170 - 0.985i)T \) |
| 7 | \( 1 + (0.793 + 0.607i)T \) |
| 11 | \( 1 + (0.348 + 0.937i)T \) |
| 13 | \( 1 + (0.662 - 0.748i)T \) |
| 17 | \( 1 + (-0.994 + 0.108i)T \) |
| 19 | \( 1 + (0.828 - 0.560i)T \) |
| 23 | \( 1 + (-0.970 + 0.240i)T \) |
| 29 | \( 1 + (-0.630 - 0.775i)T \) |
| 31 | \( 1 + (-0.154 + 0.988i)T \) |
| 37 | \( 1 + (-0.0293 + 0.999i)T \) |
| 41 | \( 1 + (-0.637 + 0.770i)T \) |
| 43 | \( 1 + (0.962 - 0.272i)T \) |
| 47 | \( 1 + (-0.440 - 0.897i)T \) |
| 53 | \( 1 + (-0.0627 - 0.998i)T \) |
| 59 | \( 1 + (-0.964 + 0.264i)T \) |
| 61 | \( 1 + (-0.832 + 0.553i)T \) |
| 67 | \( 1 + (0.0544 + 0.998i)T \) |
| 71 | \( 1 + (0.0125 - 0.999i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.828 - 0.560i)T \) |
| 83 | \( 1 + (-0.996 + 0.0836i)T \) |
| 89 | \( 1 + (0.244 + 0.969i)T \) |
| 97 | \( 1 + (-0.923 - 0.383i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.5275590638266116085247189033, −16.853112306898446041758378957558, −16.24288141786085253775640096887, −15.51093705607655844480301381421, −14.37377569918463316847189383293, −14.04195420111876016217847632196, −13.731055553265066849636707908776, −12.8609696273378984799454063172, −11.92933497722132304870036260467, −11.28826313710369327318907678822, −11.03142371877052301736368290871, −10.43712047792844126586465119536, −9.345783616360956976298244093892, −8.62118034009616644012578024574, −7.67505568426952338290452052678, −7.42701567699597510921255634323, −6.473300673355067416682055813581, −6.09589693674193611506656198530, −5.38930972004939482140137493788, −4.29544635944986042733358108029, −3.69544091138861338400628963655, −2.709333575218245254481516282944, −1.82595864714656543832782973368, −1.351639425924891305684426160, −0.11585743771102291046928498312,
1.172191998751204645280823835783, 1.8267754544384494958092272875, 2.86383709983941103995129094795, 3.89435403311247613804960239480, 4.5191159869555914088553882658, 5.04782925797359745439757877245, 5.65731467164572445713212023236, 6.29382968049393475699156115469, 7.30972791153757243846350439438, 8.29221462536843742494972936523, 8.70040655211409681119338580241, 9.45047209763687983718835989380, 9.951960311774109838179915243429, 10.774058196315089492592868401878, 11.61695563687267565101974277125, 11.86070780594456297453114781454, 12.67059749157500213665452374463, 13.39189579165867474896130757271, 14.13953465100899868418421788095, 15.102022272544298401145455896737, 15.42853988224588059799825533608, 15.96564718149305493902518849169, 16.734294171328609819114864230537, 17.46512265340447350198566650592, 17.862748128822316900145707021024