L(s) = 1 | + (−0.173 − 0.300i)7-s + (1.26 − 0.223i)13-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)25-s + (0.592 − 0.342i)31-s + 0.684i·37-s + (1.43 + 1.20i)43-s + (0.439 − 0.761i)49-s + (−1.17 + 0.984i)61-s + (0.673 − 1.85i)67-s + (0.266 − 1.50i)73-s + (1.93 + 0.342i)79-s + (−0.286 − 0.342i)91-s + (−0.592 − 1.62i)97-s + (1.11 + 0.642i)103-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.300i)7-s + (1.26 − 0.223i)13-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)25-s + (0.592 − 0.342i)31-s + 0.684i·37-s + (1.43 + 1.20i)43-s + (0.439 − 0.761i)49-s + (−1.17 + 0.984i)61-s + (0.673 − 1.85i)67-s + (0.266 − 1.50i)73-s + (1.93 + 0.342i)79-s + (−0.286 − 0.342i)91-s + (−0.592 − 1.62i)97-s + (1.11 + 0.642i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.238151335\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.238151335\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 0.684iT - T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.673 + 1.85i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (-1.93 - 0.342i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
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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
−8.908112158284956979818283063001, −8.197303742011951452719986119121, −7.51586210840079767263019905156, −6.40593957024354423929227474105, −6.16742368209472320779461116499, −4.94048917319105621652434294404, −4.19131193662584885211295872997, −3.30504962496349480154310217232, −2.29972583153627645105063306829, −0.927118342063271800821952834942,
1.30633350690278709621312099625, 2.42584975755551579093993924770, 3.57434527778619797696092959446, 4.16317568230237485694964151403, 5.38180426893978093776806788669, 5.97539070574982010648247029506, 6.72745823503081153699784536783, 7.62112511478840128215323505419, 8.404389681883510962042362305674, 9.023253934307894279247950577312