L(s) = 1 | − i·2-s + (0.535 − 0.309i)5-s + (0.809 − 1.40i)7-s − i·8-s + (−0.309 − 0.535i)10-s + 0.618i·11-s + (−0.309 + 0.535i)13-s + (−1.40 − 0.809i)14-s − 16-s + (−0.535 − 0.309i)17-s + (0.809 + 1.40i)19-s + 0.618·22-s + (−1.40 + 0.809i)23-s + (−0.309 + 0.535i)25-s + (0.535 + 0.309i)26-s + ⋯ |
L(s) = 1 | − i·2-s + (0.535 − 0.309i)5-s + (0.809 − 1.40i)7-s − i·8-s + (−0.309 − 0.535i)10-s + 0.618i·11-s + (−0.309 + 0.535i)13-s + (−1.40 − 0.809i)14-s − 16-s + (−0.535 − 0.309i)17-s + (0.809 + 1.40i)19-s + 0.618·22-s + (−1.40 + 0.809i)23-s + (−0.309 + 0.535i)25-s + (0.535 + 0.309i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.331241817\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.331241817\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + iT - T^{2} \) |
| 5 | \( 1 + (-0.535 + 0.309i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - 0.618iT - T^{2} \) |
| 13 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.535 + 0.309i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.40 - 0.809i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 1.61iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1.40 + 0.809i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + 1.61iT - T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−9.968612722283327105621416036071, −9.381254111946807268662702542557, −7.978038375883224155212405493478, −7.37510001251980287147023519306, −6.49946761533100789553331648770, −5.27079735955058880956072836470, −4.25020745765000525940001492280, −3.58786312166263280871750927507, −2.01174394344928704448951503952, −1.41155860770899583969212521682,
2.14002882726016571885559672979, 2.74083826748568944068672135663, 4.56891060241296614409543295324, 5.55619869823116604818805016192, 5.90584892571361168437670063141, 6.82335290377932177831457788952, 7.78038482994683114253657554239, 8.525390669854641762655398807733, 9.025826589475942406781519854705, 10.20630861160793010334705133632