L(s) = 1 | + (−0.866 + 0.5i)2-s − 3-s + (0.499 − 0.866i)4-s + (−0.620 − 0.358i)5-s + (0.866 − 0.5i)6-s + (−2.52 + 0.792i)7-s + 0.999i·8-s + 9-s + 0.716·10-s + 2.57i·11-s + (−0.499 + 0.866i)12-s + (3.57 + 0.467i)13-s + (1.78 − 1.94i)14-s + (0.620 + 0.358i)15-s + (−0.5 − 0.866i)16-s + (−0.0124 + 0.0215i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s − 0.577·3-s + (0.249 − 0.433i)4-s + (−0.277 − 0.160i)5-s + (0.353 − 0.204i)6-s + (−0.954 + 0.299i)7-s + 0.353i·8-s + 0.333·9-s + 0.226·10-s + 0.775i·11-s + (−0.144 + 0.249i)12-s + (0.991 + 0.129i)13-s + (0.478 − 0.520i)14-s + (0.160 + 0.0924i)15-s + (−0.125 − 0.216i)16-s + (−0.00301 + 0.00521i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.386623 - 0.300174i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.386623 - 0.300174i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (2.52 - 0.792i)T \) |
| 13 | \( 1 + (-3.57 - 0.467i)T \) |
good | 5 | \( 1 + (0.620 + 0.358i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 2.57iT - 11T^{2} \) |
| 17 | \( 1 + (0.0124 - 0.0215i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + 6.91iT - 19T^{2} \) |
| 23 | \( 1 + (4.69 + 8.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.77 - 4.80i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.92 + 2.84i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.61 + 2.08i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.54 - 2.62i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.37 + 11.0i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.32 + 2.49i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.72 + 8.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.95 - 1.12i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 0.652T + 61T^{2} \) |
| 67 | \( 1 - 1.87iT - 67T^{2} \) |
| 71 | \( 1 + (-7.52 + 4.34i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.7 + 7.94i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.194 - 0.336i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.85iT - 83T^{2} \) |
| 89 | \( 1 + (12.8 - 7.41i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.78 - 1.03i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.49433924956102847681451495584, −9.723071494778264158196424813907, −8.880542824122905877918085512310, −8.007058793490598748977627270864, −6.69901597357407807373546520424, −6.42539998246127625532138680651, −5.13613713490860522083369017681, −4.00778644319041437879965703138, −2.34672975396414332119022067909, −0.40136022021364167954458068201,
1.33549587749925029921506640631, 3.27587723381757939287537078604, 3.94750135390532725208965279181, 5.81356107340437270916055985109, 6.29549603791696208028113090485, 7.59823588144733443758218997244, 8.233064436222287383574969527789, 9.573953975753391942684553718655, 9.994504890705755160413314227722, 11.09314926418291112524709854376