L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.204 − 0.171i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s − 0.267·6-s + (−3.03 − 1.10i)7-s + (0.500 − 0.866i)8-s + (−0.508 + 2.88i)9-s + (0.5 + 0.866i)10-s + (−0.746 + 1.29i)11-s + (0.204 + 0.171i)12-s + (0.0469 + 0.266i)13-s + (1.61 + 2.79i)14-s + (−0.251 + 0.0914i)15-s + (−0.939 + 0.342i)16-s + (−0.420 + 2.38i)17-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.118 − 0.0992i)3-s + (0.0868 + 0.492i)4-s + (−0.420 − 0.152i)5-s − 0.109·6-s + (−1.14 − 0.417i)7-s + (0.176 − 0.306i)8-s + (−0.169 + 0.961i)9-s + (0.158 + 0.273i)10-s + (−0.225 + 0.390i)11-s + (0.0591 + 0.0496i)12-s + (0.0130 + 0.0738i)13-s + (0.431 + 0.748i)14-s + (−0.0649 + 0.0236i)15-s + (−0.234 + 0.0855i)16-s + (−0.101 + 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.152312 + 0.240288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.152312 + 0.240288i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 37 | \( 1 + (5.96 - 1.21i)T \) |
good | 3 | \( 1 + (-0.204 + 0.171i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (3.03 + 1.10i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.746 - 1.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0469 - 0.266i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.420 - 2.38i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (5.39 - 4.52i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (1.46 + 2.53i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0380 - 0.0659i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 41 | \( 1 + (-0.903 - 5.12i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 + (-1.45 - 2.51i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.50 + 2.00i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-4.61 + 1.68i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.0607 - 0.344i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.52 + 2.37i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.03 + 2.54i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 0.00494T + 73T^{2} \) |
| 79 | \( 1 + (14.6 + 5.34i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.80 + 15.9i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.04 + 1.11i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (5.03 + 8.72i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.61003584121297575483169777945, −10.43912642162415306967557336075, −10.19403432484772055731347513382, −8.869613478946451012293936388941, −8.103943967011308059451997200956, −7.17831830026734248946041067643, −6.10141219524105229797326624250, −4.49908468381543744642722636729, −3.41434362531056140742643895667, −2.01369836985851388069966515242,
0.21172068011346814346779111334, 2.71458353552684314818090188296, 3.89170583976855795560892704621, 5.50903321408053992361209237034, 6.50374470707694472926065055537, 7.18007006548003921708984219397, 8.567694877643010426444342305371, 9.101251853644916540930383908021, 9.988357385870321937167806632119, 10.98688641418369005389463373087