L(s) = 1 | + (1.89 − 1.09i)2-s + (−0.5 − 0.866i)3-s + (1.39 − 2.41i)4-s + (−1.89 − 1.09i)6-s + (−1.5 − 0.866i)7-s − 1.73i·8-s + (1 − 1.73i)9-s + (2.29 − 1.32i)11-s − 2.79·12-s + (−1 − 3.46i)13-s − 3.79·14-s + (0.895 + 1.55i)16-s + (−2.29 + 3.96i)17-s − 4.37i·18-s + (1.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (1.34 − 0.773i)2-s + (−0.288 − 0.499i)3-s + (0.697 − 1.20i)4-s + (−0.773 − 0.446i)6-s + (−0.566 − 0.327i)7-s − 0.612i·8-s + (0.333 − 0.577i)9-s + (0.690 − 0.398i)11-s − 0.805·12-s + (−0.277 − 0.960i)13-s − 1.01·14-s + (0.223 + 0.387i)16-s + (−0.555 + 0.962i)17-s − 1.03i·18-s + (0.344 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42502 - 1.84485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42502 - 1.84485i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 + (-1.89 + 1.09i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.29 + 1.32i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.29 - 3.96i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.29 - 3.96i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.29 - 3.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.66iT - 31T^{2} \) |
| 37 | \( 1 + (-6.87 + 3.96i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.29 - 1.32i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.708 - 1.22i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8.75iT - 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 + (-2.91 - 1.68i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.29 + 9.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.8 - 7.43i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.08 + 1.77i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 + (3.70 - 2.14i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.87 - 2.23i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.63137120791725017447186829873, −10.75124978222256162900707369695, −9.854332240901221920902920579878, −8.539906117571922625388071821822, −7.06115056004437633628981360395, −6.22988354968613841483745737696, −5.26841311765211245210416347601, −3.90678491358934518309408138495, −3.14910051132875279422328253182, −1.37277825334865716856496814071,
2.66257937384224182243757120324, 4.28274589033271267660801127895, 4.66291095058155772363953286685, 5.93994624560785862265356508317, 6.74404776009806882788166735013, 7.62004421921352067170964512435, 9.238271985888426153310402988148, 9.896667925780629804186978014448, 11.35197723284907706685286184827, 11.97387849940462873845815874447