L(s) = 1 | + (−2.59 + 0.843i)2-s + (4.40 − 3.19i)4-s + (1.55 + 2.14i)7-s + (−5.52 + 7.60i)8-s + (0.927 + 2.85i)9-s + (3.13 − 1.08i)11-s + (−5.83 − 4.24i)14-s + (4.55 − 14.0i)16-s + (−4.81 − 6.62i)18-s + (−7.21 + 5.45i)22-s + 3.36·23-s + (−4.04 − 2.93i)25-s + (13.6 + 4.45i)28-s + (−3.04 − 4.18i)29-s + 21.4i·32-s + ⋯ |
L(s) = 1 | + (−1.83 + 0.596i)2-s + (2.20 − 1.59i)4-s + (0.587 + 0.809i)7-s + (−1.95 + 2.68i)8-s + (0.309 + 0.951i)9-s + (0.945 − 0.326i)11-s + (−1.56 − 1.13i)14-s + (1.13 − 3.50i)16-s + (−1.13 − 1.56i)18-s + (−1.53 + 1.16i)22-s + 0.700·23-s + (−0.809 − 0.587i)25-s + (2.58 + 0.841i)28-s + (−0.564 − 0.777i)29-s + 3.78i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.425226 + 0.227748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.425226 + 0.227748i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.55 - 2.14i)T \) |
| 11 | \( 1 + (-3.13 + 1.08i)T \) |
good | 2 | \( 1 + (2.59 - 0.843i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 + (3.04 + 4.18i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (8.95 - 6.50i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 13.0iT - 43T^{2} \) |
| 47 | \( 1 + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.15 + 6.63i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + (-0.0272 + 0.0839i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.57 - 0.835i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (78.4 - 57.0i)T^{2} \) |
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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
−15.15118592944148043678640643998, −13.99234998221701628826412520131, −11.91495619063024775539342888904, −11.05899630457667658923997782251, −9.941190405211439348674627653014, −8.824045289957561167869707846372, −8.048198799348857011334735753900, −6.81621386434999215812144164943, −5.46482451547966621051434341265, −1.94007870270729304110028665313,
1.39278900581271508679553308498, 3.68223452809524052525377654786, 6.71377062309938648770693824469, 7.57973266816025702740969860656, 8.937913606254475678370381565684, 9.701961755082570657302423710435, 10.85444620144125860015404165417, 11.68767726605641525773112976917, 12.73275704813754968382814225044, 14.57352531423124203427585643286