L(s) = 1 | + (3.91 − 3.91i)3-s + (−4.72 + 4.72i)5-s − 45.3·7-s + 50.3i·9-s + (110. + 110. i)11-s + (157. + 157. i)13-s + 36.9i·15-s − 378.·17-s + (203. − 203. i)19-s + (−177. + 177. i)21-s + 740.·23-s + 580. i·25-s + (514. + 514. i)27-s + (−82.6 − 82.6i)29-s + 286. i·31-s + ⋯ |
L(s) = 1 | + (0.434 − 0.434i)3-s + (−0.188 + 0.188i)5-s − 0.925·7-s + 0.621i·9-s + (0.910 + 0.910i)11-s + (0.929 + 0.929i)13-s + 0.164i·15-s − 1.31·17-s + (0.562 − 0.562i)19-s + (−0.402 + 0.402i)21-s + 1.39·23-s + 0.928i·25-s + (0.705 + 0.705i)27-s + (−0.0982 − 0.0982i)29-s + 0.297i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.42042 + 0.866554i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42042 + 0.866554i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-3.91 + 3.91i)T - 81iT^{2} \) |
| 5 | \( 1 + (4.72 - 4.72i)T - 625iT^{2} \) |
| 7 | \( 1 + 45.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-110. - 110. i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (-157. - 157. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 378.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (-203. + 203. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 - 740.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (82.6 + 82.6i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 - 286. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (1.47e3 - 1.47e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.30e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (366. + 366. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 751. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-1.92e3 + 1.92e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (1.35e3 + 1.35e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (-1.83e3 - 1.83e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (-2.20e3 + 2.20e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 8.97e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.35e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 2.86e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (1.03e3 - 1.03e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 5.17e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.53e3T + 8.85e7T^{2} \) |
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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
−13.14254694652712202741629313997, −11.78707458637571215598827857172, −10.86573675101625824664391575253, −9.432067355600792142744779327915, −8.728018262153114982260286100415, −7.12523026654152803879928616885, −6.62238091740757854082467432146, −4.69836546758112868691661178699, −3.23216144555519374258734826568, −1.65936930420181572104160360815,
0.69591804884145730657163591976, 3.13612307607975077017458026032, 3.97656530663399418593347847996, 5.84377862621651189388376926560, 6.84498951677803116611357708637, 8.598912369640877782732140624776, 9.077075110646706086501720892937, 10.30445247826839011223181517322, 11.38864488515808634496167020501, 12.56034041949247003301341775031