| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.748 − 1.56i)3-s + (0.173 − 0.984i)4-s + (−0.262 + 0.0462i)5-s + (−1.57 − 0.715i)6-s + (0.604 − 1.04i)7-s + (−0.500 − 0.866i)8-s + (−1.88 + 2.33i)9-s + (−0.171 + 0.204i)10-s + (2.03 − 1.17i)11-s + (−1.66 + 0.465i)12-s + (1.01 + 2.79i)13-s + (−0.209 − 1.19i)14-s + (0.268 + 0.375i)15-s + (−0.939 − 0.342i)16-s + (0.576 + 0.687i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.431 − 0.901i)3-s + (0.0868 − 0.492i)4-s + (−0.117 + 0.0206i)5-s + (−0.643 − 0.292i)6-s + (0.228 − 0.395i)7-s + (−0.176 − 0.306i)8-s + (−0.626 + 0.779i)9-s + (−0.0541 + 0.0645i)10-s + (0.613 − 0.354i)11-s + (−0.481 + 0.134i)12-s + (0.282 + 0.775i)13-s + (−0.0561 − 0.318i)14-s + (0.0693 + 0.0968i)15-s + (−0.234 − 0.0855i)16-s + (0.139 + 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0447 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0447 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.868542 - 0.830480i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.868542 - 0.830480i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.748 + 1.56i)T \) |
| 19 | \( 1 + (-1.97 - 3.88i)T \) |
| good | 5 | \( 1 + (0.262 - 0.0462i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.604 + 1.04i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.03 + 1.17i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 2.79i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.576 - 0.687i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-5.53 - 0.976i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.92 - 1.61i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (8.98 + 5.18i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.95iT - 37T^{2} \) |
| 41 | \( 1 + (10.4 + 3.79i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.834 + 4.73i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.24 + 1.48i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.998 + 5.66i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (9.78 - 8.20i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.153 - 0.871i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.28 + 3.91i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.64 + 9.33i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.320 - 0.116i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.33 + 6.42i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 7.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (11.5 - 4.20i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.88 - 4.62i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−13.39976255831573539695952269935, −12.18774325916579786362047008842, −11.52412195740575725958189644906, −10.61440687050636976654902748745, −9.079852538565870608985083872389, −7.62576524441450155777331232352, −6.52070668291588903260328630941, −5.33447486694201472616081276192, −3.68612650456436155842637875513, −1.59828000032087878682008238102,
3.30090229005543295211359344759, 4.71385684995951048204467241725, 5.67508273471761897308477654109, 6.99642534668812926547742260358, 8.529748641098557650353512023747, 9.552822670248169689765273095189, 10.89850411279937541229342079768, 11.76702406795626933515311858577, 12.76468065835107804934146088552, 14.07397398824696815640733631334
