| L(s) = 1 | + (−0.595 − 1.16i)2-s + (−0.981 − 1.42i)3-s + (0.163 − 0.225i)4-s + (−0.0620 − 0.391i)5-s + (−1.08 + 1.99i)6-s + (0.908 − 0.775i)7-s + (−2.95 − 0.467i)8-s + (−1.07 + 2.80i)9-s + (−0.420 + 0.305i)10-s + (−1.43 − 0.881i)11-s + (−0.482 − 0.0125i)12-s + (−1.10 + 0.0866i)13-s + (−1.44 − 0.599i)14-s + (−0.498 + 0.472i)15-s + (1.03 + 3.20i)16-s + (7.17 − 1.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.421 − 0.826i)2-s + (−0.566 − 0.824i)3-s + (0.0818 − 0.112i)4-s + (−0.0277 − 0.175i)5-s + (−0.442 + 0.815i)6-s + (0.343 − 0.293i)7-s + (−1.04 − 0.165i)8-s + (−0.358 + 0.933i)9-s + (−0.133 + 0.0967i)10-s + (−0.433 − 0.265i)11-s + (−0.139 − 0.00363i)12-s + (−0.305 + 0.0240i)13-s + (−0.387 − 0.160i)14-s + (−0.128 + 0.122i)15-s + (0.259 + 0.800i)16-s + (1.73 − 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.166726 - 0.703891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.166726 - 0.703891i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.981 + 1.42i)T \) |
| 41 | \( 1 + (4.37 - 4.67i)T \) |
| good | 2 | \( 1 + (0.595 + 1.16i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.0620 + 0.391i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-0.908 + 0.775i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (1.43 + 0.881i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (1.10 - 0.0866i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (-7.17 + 1.72i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (2.37 + 0.186i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-1.46 + 4.51i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.31 - 1.03i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-0.811 - 1.11i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.82 + 2.05i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-5.92 + 3.01i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-4.59 + 5.38i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (1.81 - 7.56i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (5.24 + 1.70i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.35 - 6.57i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (4.71 - 2.88i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (1.06 - 1.73i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (5.55 + 5.55i)T + 73iT^{2} \) |
| 79 | \( 1 + (-15.0 + 6.24i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 9.90iT - 83T^{2} \) |
| 89 | \( 1 + (-8.68 - 10.1i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (0.233 + 0.380i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−12.45448315000374491391551758774, −12.08745123036444237154391301949, −10.82176107401890339584471458756, −10.31484658582479389261422161426, −8.781899148125163958962356737855, −7.59156631022047424387353413947, −6.31601455086833557089701665823, −5.07608843611352966399684142044, −2.73430128465325286329426775803, −1.01534027012793796719248153920,
3.25718646892614420267333796144, 5.07508336688552301324608352188, 6.09220285851701597501828186976, 7.38639843412121837590758031437, 8.465937961822258393261468745778, 9.609693120425526732245879467741, 10.63473246895134829194423919532, 11.80954495261071277301372652568, 12.53925928336204728454052010876, 14.39061177813828632893598314476
