L(s) = 1 | + (0.334 − 0.334i)5-s + 4.55i·7-s + (−2.47 + 2.47i)11-s + (−0.0594 − 0.0594i)13-s − 3.61·17-s + (−2.55 − 2.55i)19-s + 2.82i·23-s + 4.77i·25-s + (5.16 + 5.16i)29-s + 0.557·31-s + (1.52 + 1.52i)35-s + (4.38 − 4.38i)37-s + 9.27i·41-s + (1.61 − 1.61i)43-s + 2.82·47-s + ⋯ |
L(s) = 1 | + (0.149 − 0.149i)5-s + 1.72i·7-s + (−0.745 + 0.745i)11-s + (−0.0164 − 0.0164i)13-s − 0.877·17-s + (−0.586 − 0.586i)19-s + 0.589i·23-s + 0.955i·25-s + (0.958 + 0.958i)29-s + 0.100·31-s + (0.258 + 0.258i)35-s + (0.721 − 0.721i)37-s + 1.44i·41-s + (0.245 − 0.245i)43-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.749396 + 0.875876i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.749396 + 0.875876i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.334 + 0.334i)T - 5iT^{2} \) |
| 7 | \( 1 - 4.55iT - 7T^{2} \) |
| 11 | \( 1 + (2.47 - 2.47i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.0594 + 0.0594i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + (2.55 + 2.55i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-5.16 - 5.16i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 + (-4.38 + 4.38i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.27iT - 41T^{2} \) |
| 43 | \( 1 + (-1.61 + 1.61i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-0.493 + 0.493i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4 + 4i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.72 - 2.72i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.77 + 3.77i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.11iT - 71T^{2} \) |
| 73 | \( 1 + 0.541iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (10.6 + 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−11.08034245134727140914536845127, −9.968411323550486801781325375095, −9.095323739909218693625269818247, −8.554716027323699469993891150312, −7.41643966386611745821710211678, −6.33225266671685503596920243707, −5.39115907916154851700983563375, −4.62581080167913373127355882037, −2.90896615334330355947730813311, −2.01189740602251190856633781880,
0.63536017239316297815878463937, 2.49055164261047814902178716652, 3.88201608724888892725686382369, 4.64659558221367702359858444890, 6.08446683422302200754878188379, 6.84341561122601087467685697858, 7.86388917682531788350743479826, 8.537752125981318735663594643532, 9.902236988993900297219366426487, 10.53674204917505515465211742216