L(s) = 1 | + (0.173 + 0.984i)5-s + (−0.642 − 0.766i)9-s + (0.642 − 0.766i)13-s + (1.50 − 1.26i)17-s + (−0.939 + 0.342i)25-s + (0.0451 − 0.168i)29-s + (0.766 + 0.642i)37-s + (1.26 − 1.50i)41-s + (0.642 − 0.766i)45-s + (−0.342 + 0.939i)49-s + (−1.10 + 1.58i)53-s + (1.98 − 0.173i)61-s + (0.866 + 0.5i)65-s + (−0.366 + 0.366i)73-s + (−0.173 + 0.984i)81-s + ⋯ |
L(s) = 1 | + (0.173 + 0.984i)5-s + (−0.642 − 0.766i)9-s + (0.642 − 0.766i)13-s + (1.50 − 1.26i)17-s + (−0.939 + 0.342i)25-s + (0.0451 − 0.168i)29-s + (0.766 + 0.642i)37-s + (1.26 − 1.50i)41-s + (0.642 − 0.766i)45-s + (−0.342 + 0.939i)49-s + (−1.10 + 1.58i)53-s + (1.98 − 0.173i)61-s + (0.866 + 0.5i)65-s + (−0.366 + 0.366i)73-s + (−0.173 + 0.984i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.294630369\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.294630369\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 37 | \( 1 + (-0.766 - 0.642i)T \) |
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 | \( 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-1.50 + 1.26i)T + (0.173 - 0.984i)T^{2} \) |
| 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 | \( 1 + (1.10 - 1.58i)T + (-0.342 - 0.939i)T^{2} \) |
| 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 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.060963961619343071342139498892, −7.971229115965478190288429665220, −7.50700217127506128082012931581, −6.56953246452095243172121574005, −5.89091189992771669563637929631, −5.32334028108498393853834670827, −3.97493441220672547057930830888, −3.14912854134855904873253929217, −2.63866933735507380083447478737, −0.982158062475347556990028231859,
1.22965574329752461575909540601, 2.15915779752384400567245383998, 3.44632894779437287889833616757, 4.27072469917316774192959250035, 5.17602365526183877746685428889, 5.79583919186843143512697013782, 6.49273220146175649688765192584, 7.78302975465680027591967839312, 8.163525693418598977217471321860, 8.826752414426866056124977500912