L(s) = 1 | + (−0.866 − 0.5i)5-s + (0.642 + 0.766i)9-s + (−0.766 − 0.642i)13-s + (1.26 + 1.50i)17-s + (0.499 + 0.866i)25-s + (−0.0451 + 0.168i)29-s + (−0.642 + 0.766i)37-s + (1.26 − 1.50i)41-s + (−0.173 − 0.984i)45-s + (0.342 − 0.939i)49-s + (1.58 + 1.10i)53-s + (1.98 − 0.173i)61-s + (0.342 + 0.939i)65-s + (0.366 + 0.366i)73-s + (−0.173 + 0.984i)81-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)5-s + (0.642 + 0.766i)9-s + (−0.766 − 0.642i)13-s + (1.26 + 1.50i)17-s + (0.499 + 0.866i)25-s + (−0.0451 + 0.168i)29-s + (−0.642 + 0.766i)37-s + (1.26 − 1.50i)41-s + (−0.173 − 0.984i)45-s + (0.342 − 0.939i)49-s + (1.58 + 1.10i)53-s + (1.98 − 0.173i)61-s + (0.342 + 0.939i)65-s + (0.366 + 0.366i)73-s + (−0.173 + 0.984i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.085001520\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.085001520\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + (0.642 - 0.766i)T \) |
good | 3 | \( 1 + (-0.642 - 0.766i)T^{2} \) |
| 7 | \( 1 + (-0.342 + 0.939i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \) |
| 19 | \( 1 + (0.642 + 0.766i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.0451 - 0.168i)T + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.58 - 1.10i)T + (0.342 + 0.939i)T^{2} \) |
| 59 | \( 1 + (-0.342 - 0.939i)T^{2} \) |
| 61 | \( 1 + (-1.98 + 0.173i)T + (0.984 - 0.173i)T^{2} \) |
| 67 | \( 1 + (0.342 - 0.939i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 + (0.342 - 0.939i)T^{2} \) |
| 83 | \( 1 + (0.984 + 0.173i)T^{2} \) |
| 89 | \( 1 + (0.939 - 1.34i)T + (-0.342 - 0.939i)T^{2} \) |
| 97 | \( 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.730981311544628189508576810809, −8.166858036705359501646300830719, −7.54525796386934121283286797391, −6.97478301586053881069616658993, −5.65313030594183373474951542226, −5.17934652845691664524644086021, −4.16729902143530499241922245489, −3.58567254269918340812276694345, −2.33176469493901186493835083576, −1.11586386980235295929526931719,
0.857472881014651343389061343635, 2.43862366053586383991927727477, 3.32889565441947725556516705720, 4.12324753366542655045580072435, 4.88867733112379208022036137386, 5.89078473148797291724824834700, 6.99482132733191344005019741272, 7.19521393996613788646677435716, 7.974342128062983156712790418329, 8.957573676870031656535419002574