L(s) = 1 | + (−5.52 + 5.52i)5-s − 7.79i·7-s + (−2.48 + 2.48i)11-s + (−13.3 + 13.3i)13-s + 5.86i·17-s + (18.5 − 18.5i)19-s − 34.3·23-s − 36.0i·25-s + (21.4 + 21.4i)29-s + 30.6·31-s + (43.0 + 43.0i)35-s + (−30.3 − 30.3i)37-s − 3.12·41-s + (−9.94 − 9.94i)43-s − 38.4i·47-s + ⋯ |
L(s) = 1 | + (−1.10 + 1.10i)5-s − 1.11i·7-s + (−0.225 + 0.225i)11-s + (−1.02 + 1.02i)13-s + 0.344i·17-s + (0.977 − 0.977i)19-s − 1.49·23-s − 1.44i·25-s + (0.740 + 0.740i)29-s + 0.987·31-s + (1.23 + 1.23i)35-s + (−0.819 − 0.819i)37-s − 0.0762·41-s + (−0.231 − 0.231i)43-s − 0.817i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9643148856\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9643148856\) |
\(L(2)\) |
|
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 | \( 1 + (5.52 - 5.52i)T - 25iT^{2} \) |
| 7 | \( 1 + 7.79iT - 49T^{2} \) |
| 11 | \( 1 + (2.48 - 2.48i)T - 121iT^{2} \) |
| 13 | \( 1 + (13.3 - 13.3i)T - 169iT^{2} \) |
| 17 | \( 1 - 5.86iT - 289T^{2} \) |
| 19 | \( 1 + (-18.5 + 18.5i)T - 361iT^{2} \) |
| 23 | \( 1 + 34.3T + 529T^{2} \) |
| 29 | \( 1 + (-21.4 - 21.4i)T + 841iT^{2} \) |
| 31 | \( 1 - 30.6T + 961T^{2} \) |
| 37 | \( 1 + (30.3 + 30.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 3.12T + 1.68e3T^{2} \) |
| 43 | \( 1 + (9.94 + 9.94i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-61.1 + 61.1i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (2.98 - 2.98i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (3.88 - 3.88i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (47.0 - 47.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 97.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 96.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-88.8 - 88.8i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 54.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 5.00T + 9.40e3T^{2} \) |
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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
−9.766422940560085221314667730918, −8.530887194097626856355065280204, −7.56479133318890460960780358871, −7.13771379986183518824675775515, −6.54583357562802933925810506373, −4.99354870745275127398610052688, −4.12747608387531972773508366368, −3.39509649780914753329168443104, −2.21660440883293833492688453325, −0.39961744469248988087880252159,
0.836745065276158233240050955381, 2.44680961002433006952159882146, 3.51557373433193329794559473747, 4.67163991249912967350646090596, 5.31216228113475405764850832119, 6.15829871353521172485223810842, 7.69558405322020091211822580934, 7.986027167031818606875499744599, 8.727558699487891531058668888037, 9.669095102433463690849118088672