L(s) = 1 | + (−0.774 + 0.633i)2-s + (0.809 − 0.587i)3-s + (0.198 − 0.980i)4-s + (−0.254 + 0.967i)6-s + (−0.0855 − 0.996i)7-s + (0.466 + 0.884i)8-s + (0.309 − 0.951i)9-s + (−0.415 − 0.909i)12-s + (−0.993 + 0.113i)13-s + (0.696 + 0.717i)14-s + (−0.921 − 0.389i)16-s + (0.564 + 0.825i)17-s + (0.362 + 0.931i)18-s + (0.941 − 0.336i)19-s + (−0.654 − 0.755i)21-s + ⋯ |
L(s) = 1 | + (−0.774 + 0.633i)2-s + (0.809 − 0.587i)3-s + (0.198 − 0.980i)4-s + (−0.254 + 0.967i)6-s + (−0.0855 − 0.996i)7-s + (0.466 + 0.884i)8-s + (0.309 − 0.951i)9-s + (−0.415 − 0.909i)12-s + (−0.993 + 0.113i)13-s + (0.696 + 0.717i)14-s + (−0.921 − 0.389i)16-s + (0.564 + 0.825i)17-s + (0.362 + 0.931i)18-s + (0.941 − 0.336i)19-s + (−0.654 − 0.755i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9374805910 - 0.6737096587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9374805910 - 0.6737096587i\) |
\(L(1)\) |
\(\approx\) |
\(0.9291378244 - 0.1878593590i\) |
\(L(1)\) |
\(\approx\) |
\(0.9291378244 - 0.1878593590i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.774 + 0.633i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.0855 - 0.996i)T \) |
| 13 | \( 1 + (-0.993 + 0.113i)T \) |
| 17 | \( 1 + (0.564 + 0.825i)T \) |
| 19 | \( 1 + (0.941 - 0.336i)T \) |
| 23 | \( 1 + (0.654 - 0.755i)T \) |
| 29 | \( 1 + (0.610 + 0.791i)T \) |
| 31 | \( 1 + (-0.736 - 0.676i)T \) |
| 37 | \( 1 + (0.870 - 0.491i)T \) |
| 41 | \( 1 + (-0.998 + 0.0570i)T \) |
| 43 | \( 1 + (0.142 - 0.989i)T \) |
| 47 | \( 1 + (0.362 - 0.931i)T \) |
| 53 | \( 1 + (0.921 - 0.389i)T \) |
| 59 | \( 1 + (-0.998 - 0.0570i)T \) |
| 61 | \( 1 + (0.774 + 0.633i)T \) |
| 67 | \( 1 + (-0.841 - 0.540i)T \) |
| 71 | \( 1 + (-0.0285 - 0.999i)T \) |
| 73 | \( 1 + (-0.974 + 0.226i)T \) |
| 79 | \( 1 + (-0.985 - 0.170i)T \) |
| 83 | \( 1 + (-0.516 + 0.856i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.897 - 0.441i)T \) |
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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
−22.89926121994992885198402738982, −21.94513241018131138313624961982, −21.54124656875418672101517032823, −20.63251718914185461063785716120, −19.94802469965565480301776268358, −19.131826864142085955007499590135, −18.537533807501248612598088011561, −17.56963251387405942431016904700, −16.5042227769094960565299994667, −15.83146173284772340873407552314, −14.98790563800678749475667997491, −14.00452087290416998789624956380, −12.96944477131264619340619635169, −12.03597013317798803245225095846, −11.3195797256451085838351703764, −10.0248344091241328373346320107, −9.602023079780209601276344784863, −8.837487608538448930010798468388, −7.88821976279925351740883351370, −7.179535929318774596985229259802, −5.49523952328062124875184537756, −4.48747388607456981288691009684, −3.0836268586765236371751881139, −2.75571400787889305745960125978, −1.486981461294584734261638390547,
0.714063288215434928846511136386, 1.7889491640521815739956645739, 3.00714802983680851843389512264, 4.30166865497517981107361572725, 5.55533746223551186240677070422, 6.859816389514259491046011447394, 7.2509435836501901814473338046, 8.10017658793534891011866460750, 9.01623349776520962905770408103, 9.88169850274288630068199171165, 10.611173788732379924546066378542, 11.88902931971706363187708531819, 12.9829368798733826832493138889, 13.89006285693045958822929125522, 14.58800281368278269864697876172, 15.22745969426524553194819267722, 16.48618301875944744800225543849, 17.044471818757575060850613016596, 17.996081235278960333875038764228, 18.72634810199494489828472175224, 19.660639041383588414584794766228, 20.0097580131558362669242797469, 20.89340475717132650444584541308, 22.21150133908910988748960412945, 23.40786515582517368517443517319