L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.965 + 0.258i)3-s + (0.499 + 0.866i)4-s + (2.12 − 0.707i)5-s + (0.965 + 0.258i)6-s + (−1.56 − 2.71i)7-s − 0.999i·8-s + (0.866 − 0.499i)9-s + (−2.19 − 0.448i)10-s + (−3.99 + 1.06i)11-s + (−0.707 − 0.707i)12-s + (3.53 + 0.707i)13-s + 3.13i·14-s + (−1.86 + 1.23i)15-s + (−0.5 + 0.866i)16-s + (1.66 − 6.22i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.557 + 0.149i)3-s + (0.249 + 0.433i)4-s + (0.948 − 0.316i)5-s + (0.394 + 0.105i)6-s + (−0.591 − 1.02i)7-s − 0.353i·8-s + (0.288 − 0.166i)9-s + (−0.692 − 0.141i)10-s + (−1.20 + 0.322i)11-s + (−0.204 − 0.204i)12-s + (0.980 + 0.196i)13-s + 0.836i·14-s + (−0.481 + 0.318i)15-s + (−0.125 + 0.216i)16-s + (0.404 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.494802 - 0.608907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.494802 - 0.608907i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
| 13 | \( 1 + (-3.53 - 0.707i)T \) |
good | 7 | \( 1 + (1.56 + 2.71i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.99 - 1.06i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.66 + 6.22i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.560 - 2.09i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (1.49 + 5.57i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (7.51 + 4.33i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.54 + 4.54i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.351 + 0.609i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.133 + 0.497i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (3.38 + 0.905i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 4.69T + 47T^{2} \) |
| 53 | \( 1 + (8.92 + 8.92i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9.64 - 2.58i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.20 - 7.28i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.13 - 4.12i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.47 - 1.73i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 + 10.6iT - 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + (-4.02 - 15.0i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (3.60 - 2.08i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.82343544604051152601979550225, −10.02176499126619820132752963677, −9.681816317482392442057518038380, −8.386630228987725573407598499658, −7.30024013124282137743566203611, −6.34186185968782647586929359109, −5.27319309496861308107571865781, −3.98016911780624111293283810285, −2.42246699272583060347722261145, −0.67958252510744377457143937226,
1.74687074324416179597910832543, 3.17547184407504068945141532815, 5.40648630342726445712571100611, 5.83508161365641391326765945417, 6.64745495352594858784001460473, 7.944873449732042118582683148279, 8.849361570367360828400470208709, 9.777794761076147573381940975736, 10.59741357301787682877252675438, 11.22778135910005626572008628154