L(s) = 1 | + (1 − 1.73i)3-s + (1 − 1.73i)4-s + (1.5 + 2.59i)5-s + (−0.499 − 0.866i)9-s + (−1.99 − 3.46i)12-s + 13-s + 6·15-s + (−1.99 − 3.46i)16-s + (3 − 5.19i)17-s + (3.5 + 6.06i)19-s + 6·20-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + 4.00·27-s − 9·29-s + ⋯ |
L(s) = 1 | + (0.577 − 0.999i)3-s + (0.5 − 0.866i)4-s + (0.670 + 1.16i)5-s + (−0.166 − 0.288i)9-s + (−0.577 − 0.999i)12-s + 0.277·13-s + 1.54·15-s + (−0.499 − 0.866i)16-s + (0.727 − 1.26i)17-s + (0.802 + 1.39i)19-s + 1.34·20-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.769·27-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.04520 - 1.01386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.04520 - 1.01386i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−10.20178575872346732636842172104, −9.943476455264955985501598289319, −8.666662431911456873433076355555, −7.32382781730063247368061341758, −7.15422656240831436384898795626, −6.05956903679745416831384629871, −5.34970223126175432920954939460, −3.33655919731409294451986272758, −2.35575860272932616176503063076, −1.45239219827050149743360769574,
1.72524943218544239780109093382, 3.20996845857271698451291459010, 4.00330740096828896103718494942, 5.05728078844996327807484628377, 6.06474884243324355243380018483, 7.40020227169092827934891774212, 8.307352463725448282309612326081, 9.099520615524474934016293733985, 9.547652114237072681773877091954, 10.60320733949262905995285434806