| L(s) = 1 | + (−0.229 − 1.39i)2-s + (−0.385 + 1.68i)3-s + (−1.89 + 0.639i)4-s + (−0.128 + 0.121i)5-s + (2.44 + 0.151i)6-s + (3.74 − 0.438i)7-s + (1.32 + 2.49i)8-s + (−2.70 − 1.30i)9-s + (0.199 + 0.152i)10-s + (1.03 − 0.308i)11-s + (−0.348 − 3.44i)12-s + (−5.21 + 0.303i)13-s + (−1.46 − 5.13i)14-s + (−0.155 − 0.264i)15-s + (3.18 − 2.42i)16-s + (6.40 + 2.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.161 − 0.986i)2-s + (−0.222 + 0.974i)3-s + (−0.947 + 0.319i)4-s + (−0.0576 + 0.0544i)5-s + (0.998 + 0.0618i)6-s + (1.41 − 0.165i)7-s + (0.468 + 0.883i)8-s + (−0.900 − 0.434i)9-s + (0.0630 + 0.0481i)10-s + (0.310 − 0.0930i)11-s + (−0.100 − 0.994i)12-s + (−1.44 + 0.0842i)13-s + (−0.392 − 1.37i)14-s + (−0.0402 − 0.0683i)15-s + (0.795 − 0.605i)16-s + (1.55 + 0.564i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19741 + 0.233377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19741 + 0.233377i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.229 + 1.39i)T \) |
| 3 | \( 1 + (0.385 - 1.68i)T \) |
| good | 5 | \( 1 + (0.128 - 0.121i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.74 + 0.438i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-1.03 + 0.308i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (5.21 - 0.303i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-6.40 - 2.32i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.405 - 1.11i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.157 - 0.0184i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (-2.32 - 4.63i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (3.63 - 4.87i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-4.35 - 5.19i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-8.28 + 5.44i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (1.76 - 7.44i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-0.972 - 1.30i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-9.25 + 5.34i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.09 - 0.926i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (3.39 + 1.46i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (1.63 - 3.26i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (2.66 + 15.0i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.76 - 10.0i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (5.71 + 3.75i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (3.50 - 5.33i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (2.18 - 12.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.61 + 9.13i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.59136355133306287772923892216, −9.956957637140513745240711963487, −9.143134835488283092576849027295, −8.220469231008752162786641169957, −7.43265152398453054457306168706, −5.51822352423162672492506500805, −4.93404988194864801839427011556, −4.01776715140812805050553342436, −2.94432950164580645728781927539, −1.39487086152597553524525580707,
0.839978744327160996391103354394, 2.37091505583436949143654122473, 4.38988106948203621695823137800, 5.28227282122134605156788449098, 5.97756764550585750115369024862, 7.35963376044409620844706187675, 7.58058756688019250815522353895, 8.393075566603840215734260852584, 9.415809074151123626686514810268, 10.38266722691026492874549003690
