| L(s) = 1 | + (1.04 + 2.05i)2-s + (−1.72 − 0.141i)3-s + (−1.95 + 2.68i)4-s + (0.262 + 1.65i)5-s + (−1.51 − 3.69i)6-s + (−0.575 + 0.491i)7-s + (−3.01 − 0.477i)8-s + (2.95 + 0.488i)9-s + (−3.13 + 2.27i)10-s + (−2.56 − 1.57i)11-s + (3.75 − 4.36i)12-s + (0.892 − 0.0702i)13-s + (−1.61 − 0.668i)14-s + (−0.218 − 2.90i)15-s + (−0.123 − 0.381i)16-s + (6.64 − 1.59i)17-s + ⋯ |
| L(s) = 1 | + (0.740 + 1.45i)2-s + (−0.996 − 0.0817i)3-s + (−0.976 + 1.34i)4-s + (0.117 + 0.741i)5-s + (−0.619 − 1.50i)6-s + (−0.217 + 0.185i)7-s + (−1.06 − 0.168i)8-s + (0.986 + 0.162i)9-s + (−0.991 + 0.720i)10-s + (−0.772 − 0.473i)11-s + (1.08 − 1.26i)12-s + (0.247 − 0.0194i)13-s + (−0.431 − 0.178i)14-s + (−0.0564 − 0.749i)15-s + (−0.0309 − 0.0953i)16-s + (1.61 − 0.386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.371089 + 1.05494i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.371089 + 1.05494i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.141i)T \) |
| 41 | \( 1 + (2.26 + 5.99i)T \) |
| good | 2 | \( 1 + (-1.04 - 2.05i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (-0.262 - 1.65i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (0.575 - 0.491i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (2.56 + 1.57i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (-0.892 + 0.0702i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (-6.64 + 1.59i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-6.20 - 0.488i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (1.15 - 3.54i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (8.22 + 1.97i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (5.18 + 7.13i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.171 + 0.124i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-1.33 + 0.679i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-5.37 + 6.29i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (-0.608 + 2.53i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-0.108 - 0.0350i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.195 + 0.383i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (6.76 - 4.14i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-0.826 + 1.34i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-8.35 - 8.35i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.126 - 0.0525i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 2.84iT - 83T^{2} \) |
| 89 | \( 1 + (4.76 + 5.57i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (-9.09 - 14.8i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−13.90623895826917441080557630266, −13.12261176282507699917041663224, −11.99109980227111372842161255344, −10.86192014693020058744333019043, −9.651173660014814940586694113912, −7.73279135380331362327563901363, −7.17169401142203419526024024056, −5.71398250821782557556308247078, −5.49812761802651572989261667027, −3.60725303241845937096899373777,
1.30563253790787902429646789947, 3.47397906380086033085802971582, 4.90431489456312422472777191453, 5.57488025364581756874799289957, 7.47616505621975782438814217925, 9.435607489457251491667221048643, 10.26986472574674808066937605865, 11.08488439606468487768233170793, 12.25468600209857134869720976045, 12.59808223539539604845012558772
