| L(s) = 1 | + (0.968 + 1.03i)2-s + (0.987 − 1.47i)3-s + (−0.123 + 1.99i)4-s + (−3.28 + 0.653i)5-s + (2.47 − 0.413i)6-s + (0.628 − 3.15i)7-s + (−2.17 + 1.80i)8-s + (−0.0604 − 0.145i)9-s + (−3.85 − 2.75i)10-s + (−1.92 + 1.28i)11-s + (2.82 + 2.15i)12-s + (−0.0837 + 0.0837i)13-s + (3.86 − 2.41i)14-s + (−2.27 + 5.49i)15-s + (−3.96 − 0.494i)16-s + (3.85 − 1.47i)17-s + ⋯ |
| L(s) = 1 | + (0.684 + 0.728i)2-s + (0.570 − 0.853i)3-s + (−0.0619 + 0.998i)4-s + (−1.46 + 0.292i)5-s + (1.01 − 0.168i)6-s + (0.237 − 1.19i)7-s + (−0.769 + 0.638i)8-s + (−0.0201 − 0.0486i)9-s + (−1.21 − 0.870i)10-s + (−0.581 + 0.388i)11-s + (0.816 + 0.621i)12-s + (−0.0232 + 0.0232i)13-s + (1.03 − 0.644i)14-s + (−0.587 + 1.41i)15-s + (−0.992 − 0.123i)16-s + (0.934 − 0.356i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16466 + 0.269728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16466 + 0.269728i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.968 - 1.03i)T \) |
| 17 | \( 1 + (-3.85 + 1.47i)T \) |
| good | 3 | \( 1 + (-0.987 + 1.47i)T + (-1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (3.28 - 0.653i)T + (4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.628 + 3.15i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (1.92 - 1.28i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (0.0837 - 0.0837i)T - 13iT^{2} \) |
| 19 | \( 1 + (-2.00 - 0.829i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (3.36 + 5.03i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.83 - 9.24i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.139 - 0.0931i)T + (11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (1.34 + 0.897i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (2.42 + 0.483i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (2.87 - 1.19i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.01 - 3.01i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.976 + 2.35i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (4.48 + 10.8i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.18 + 5.96i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + (7.46 - 11.1i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.178 - 0.0355i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-6.42 + 4.29i)T + (30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (-1.53 + 3.71i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (1.73 + 1.73i)T + 89iT^{2} \) |
| 97 | \( 1 + (-0.851 - 4.28i)T + (-89.6 + 37.1i)T^{2} \) |
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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
−14.57774819422521879578368135374, −14.01303869639327470893485221945, −12.81852307097696208336453460842, −11.99665484863189457000942906873, −10.61109343899374521178207168993, −8.280566586495728202135331435611, −7.58035686218224120605532931249, −6.99080551670321238068050448083, −4.69639332493278525748128778871, −3.31624147506465133852523702180,
3.10162726007304877897982415020, 4.23034089822203909656165296964, 5.60141098035392782309715691123, 7.985946962286332941948927010178, 9.105015194175188520499918019374, 10.29961853481034141483061051487, 11.77963190834183693363559010782, 12.09177807209992407479559423548, 13.64132623680120969304110149381, 15.08007604658274372885806041201
