| 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
−14.07397398824696815640733631334, −12.76468065835107804934146088552, −11.76702406795626933515311858577, −10.89850411279937541229342079768, −9.552822670248169689765273095189, −8.529748641098557650353512023747, −6.99642534668812926547742260358, −5.67508273471761897308477654109, −4.71385684995951048204467241725, −3.30090229005543295211359344759,
1.59828000032087878682008238102, 3.68612650456436155842637875513, 5.33447486694201472616081276192, 6.52070668291588903260328630941, 7.62576524441450155777331232352, 9.079852538565870608985083872389, 10.61440687050636976654902748745, 11.52412195740575725958189644906, 12.18774325916579786362047008842, 13.39976255831573539695952269935
