| L(s) = 1 | + (−1.17 + 0.984i)2-s + (0.673 + 0.565i)3-s + (0.0603 − 0.342i)4-s + (−0.652 + 0.237i)5-s − 1.34·6-s + (2.20 − 0.802i)7-s + (−1.26 − 2.19i)8-s + (−0.386 − 2.19i)9-s + (0.532 − 0.921i)10-s + (−0.939 − 1.62i)11-s + (0.233 − 0.196i)12-s + (−0.213 + 1.20i)13-s + (−1.79 + 3.11i)14-s + (−0.573 − 0.208i)15-s + (4.29 + 1.56i)16-s + (1.33 + 7.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.829 + 0.696i)2-s + (0.388 + 0.326i)3-s + (0.0301 − 0.171i)4-s + (−0.291 + 0.106i)5-s − 0.550·6-s + (0.833 − 0.303i)7-s + (−0.447 − 0.775i)8-s + (−0.128 − 0.730i)9-s + (0.168 − 0.291i)10-s + (−0.283 − 0.490i)11-s + (0.0675 − 0.0566i)12-s + (−0.0590 + 0.335i)13-s + (−0.480 + 0.832i)14-s + (−0.148 − 0.0539i)15-s + (1.07 + 0.391i)16-s + (0.324 + 1.83i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.482596 + 0.299326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482596 + 0.299326i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.81 - 5.80i)T \) |
| good | 2 | \( 1 + (1.17 - 0.984i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.673 - 0.565i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (0.652 - 0.237i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-2.20 + 0.802i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.939 + 1.62i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.213 - 1.20i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.33 - 7.58i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (5.03 + 4.22i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (0.326 - 0.565i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.84 + 4.93i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.06T + 31T^{2} \) |
| 41 | \( 1 + (0.592 - 3.35i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + 8.86T + 43T^{2} \) |
| 47 | \( 1 + (-4.03 + 6.98i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.00 - 2.91i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-6.81 - 2.47i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.790 + 4.48i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.35 - 1.58i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.18 - 4.34i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 + (-7.86 + 2.86i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.30 + 7.37i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (0.992 + 0.361i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.58 + 11.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−16.92893633779070290659409549222, −15.35438887335705493399391672059, −14.91414918628655673275417365881, −13.22940552442994518818685869436, −11.66087487775198411966785679195, −10.17374367386117977858222998191, −8.734375806469152532633751430862, −7.980210796175382815612498978272, −6.39191396167823777097378674101, −3.90810374685961251293096998321,
2.24210634739823826818514096308, 5.17432866861576086112478437463, 7.69410777654472499251134127601, 8.618239404316308237274954775632, 10.07034082914931648739263334912, 11.22313384424017022859908172524, 12.31893703516718631947276246544, 13.96020786456006594368267159577, 14.92390562878720227031923802736, 16.45739076562926294404299608413
