| L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.913 + 0.406i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−2.71 − 3.01i)7-s + (−0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.913 − 0.406i)10-s + (−3.78 + 0.804i)11-s + (0.104 + 0.994i)12-s + (−0.578 + 5.49i)13-s + (−3.97 − 0.843i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (2.13 + 0.454i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.527 + 0.234i)3-s + (0.154 − 0.475i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (−1.02 − 1.14i)7-s + (−0.109 − 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.288 − 0.128i)10-s + (−1.14 + 0.242i)11-s + (0.0301 + 0.287i)12-s + (−0.160 + 1.52i)13-s + (−1.06 − 0.225i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.518 + 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0804591 + 0.138314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0804591 + 0.138314i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (5.20 - 1.96i)T \) |
| good | 7 | \( 1 + (2.71 + 3.01i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (3.78 - 0.804i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.578 - 5.49i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-2.13 - 0.454i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.0679 - 0.646i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-2.60 - 8.02i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (4.53 - 3.29i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.93 + 5.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.03 + 1.79i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.0588 + 0.559i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (6.13 + 4.45i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-7.93 + 8.81i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (4.15 - 1.85i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 + (6.75 + 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.99 - 11.1i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (9.99 - 2.12i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (16.9 + 3.60i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-3.33 - 1.48i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.72 - 8.37i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.64 + 8.15i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.35482840708885946312692355170, −9.779589165776083066475977867749, −8.991348395226161281585777259554, −7.38344980199495860138500598188, −7.05166719360408583075764232724, −5.80110090356932313298277747211, −5.01620622922363299556699278716, −4.01116766621543147865837792264, −3.34445999702157258292414335594, −1.63152407683843085696756943096,
0.06333171923508497599752646187, 2.66897828388223020307918757730, 3.11975734421388825216084918526, 4.71006202626104305849790809610, 5.74174305408469577944600529223, 5.99077714115049516766887089758, 7.15839778108740068141762502212, 7.907891611045387891133968286004, 8.770479889164066719448120804707, 9.985551916580349062863227262664
