L(s) = 1 | + (0.5 − 0.866i)5-s + (0.866 + 0.5i)9-s + (−0.5 − 0.866i)13-s + (0.133 + 0.5i)17-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.866 + 0.5i)37-s + (−0.5 − 0.133i)41-s + (0.866 − 0.499i)45-s + (0.5 + 0.866i)49-s + (1.36 − 1.36i)53-s + (−0.866 + 1.5i)61-s − 0.999·65-s − 1.73·73-s + (0.499 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (0.866 + 0.5i)9-s + (−0.5 − 0.866i)13-s + (0.133 + 0.5i)17-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.866 + 0.5i)37-s + (−0.5 − 0.133i)41-s + (0.866 − 0.499i)45-s + (0.5 + 0.866i)49-s + (1.36 − 1.36i)53-s + (−0.866 + 1.5i)61-s − 0.999·65-s − 1.73·73-s + (0.499 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.157351219\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.157351219\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + 1.73T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.26777218115655879420974208131, −9.259411859503476411887222982443, −8.381510928208263945286760205661, −7.69639521353279897320999513971, −6.68501285134507675693761931213, −5.61705041875666168611947891695, −4.91660574809820131042348097171, −4.00667444945871542675185413848, −2.54132378348475683970109012076, −1.32026222487940109306160086965,
1.69006644860720648205069248055, 2.86412678134790685549385799042, 3.95127342811784061826720536103, 4.98532173271606539023482608802, 6.10351757089718999638692786507, 6.96629908749976875864566481299, 7.33662389879960786908637803361, 8.711347172460026023548348011932, 9.514413159346275637415404016006, 10.13844502790883967598031827852