| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−1.42 − 0.124i)3-s + (0.173 + 0.984i)4-s + (−2.10 + 0.746i)5-s + (1.01 + 1.01i)6-s + (−1.15 − 2.46i)7-s + (0.500 − 0.866i)8-s + (−0.930 − 0.163i)9-s + (2.09 + 0.783i)10-s + (3.71 + 2.14i)11-s + (−0.124 − 1.42i)12-s + (−0.117 − 0.666i)13-s + (−0.705 + 2.63i)14-s + (3.10 − 0.802i)15-s + (−0.939 + 0.342i)16-s + (5.81 + 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.824 − 0.0721i)3-s + (0.0868 + 0.492i)4-s + (−0.942 + 0.333i)5-s + (0.413 + 0.413i)6-s + (−0.435 − 0.933i)7-s + (0.176 − 0.306i)8-s + (−0.310 − 0.0546i)9-s + (0.662 + 0.247i)10-s + (1.12 + 0.647i)11-s + (−0.0360 − 0.412i)12-s + (−0.0326 − 0.184i)13-s + (−0.188 + 0.703i)14-s + (0.801 − 0.207i)15-s + (−0.234 + 0.0855i)16-s + (1.41 + 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.551124 + 0.0864161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.551124 + 0.0864161i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.10 - 0.746i)T \) |
| 37 | \( 1 + (6.03 - 0.786i)T \) |
| good | 3 | \( 1 + (1.42 + 0.124i)T + (2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (1.15 + 2.46i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-3.71 - 2.14i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.117 + 0.666i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-5.81 - 1.02i)T + (15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.656 - 7.50i)T + (-18.7 - 3.29i)T^{2} \) |
| 23 | \( 1 + (-1.35 - 2.34i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.25 + 0.605i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (2.13 - 2.13i)T - 31iT^{2} \) |
| 41 | \( 1 + (-8.84 + 1.55i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 1.99T + 43T^{2} \) |
| 47 | \( 1 + (-0.904 + 3.37i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.83 - 3.92i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (-3.15 + 6.75i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-10.1 - 7.09i)T + (20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (11.0 - 5.17i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (11.6 - 9.77i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.13 + 2.13i)T + 73iT^{2} \) |
| 79 | \( 1 + (12.2 - 5.69i)T + (50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (-8.52 - 12.1i)T + (-28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-6.47 - 3.02i)T + (57.2 + 68.1i)T^{2} \) |
| 97 | \( 1 + (-5.07 + 2.92i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.49281073194569163562548816681, −10.47619623066722138456584691650, −10.00065364533159509738704732867, −8.668785823280100203783132656451, −7.59417710937174038519548714854, −6.90794561907835033481799028506, −5.76130278503875043257307773459, −4.11710153763038306022403916071, −3.37237771168583940592843991254, −1.09733167711704151819064309161,
0.64656996041080217712184204310, 3.06776730687436794339392329148, 4.67117136487549720393977315781, 5.69192468582782625380072765535, 6.50977787353893148529930146699, 7.57451175036954922099149553147, 8.863440711034586366625577005444, 9.072471699928044295417745365156, 10.55771369493002359957608305011, 11.52948993113940348566649564734
