L(s) = 1 | + (1.29 + 0.347i)2-s + (−0.170 − 0.0981i)4-s + (0.530 − 1.97i)7-s + (−2.08 − 2.08i)8-s + (0.762 − 0.440i)11-s + (−1.43 − 5.36i)13-s + (1.37 − 2.38i)14-s + (−1.78 − 3.09i)16-s + (1.13 − 1.13i)17-s + 1.52i·19-s + (1.14 − 0.305i)22-s + (1.53 − 0.410i)23-s − 7.46i·26-s + (−0.284 + 0.284i)28-s + (0.796 + 1.37i)29-s + ⋯ |
L(s) = 1 | + (0.917 + 0.245i)2-s + (−0.0850 − 0.0490i)4-s + (0.200 − 0.747i)7-s + (−0.737 − 0.737i)8-s + (0.229 − 0.132i)11-s + (−0.398 − 1.48i)13-s + (0.367 − 0.636i)14-s + (−0.446 − 0.772i)16-s + (0.275 − 0.275i)17-s + 0.349i·19-s + (0.243 − 0.0652i)22-s + (0.319 − 0.0856i)23-s − 1.46i·26-s + (−0.0537 + 0.0537i)28-s + (0.147 + 0.256i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59322 - 1.08157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59322 - 1.08157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.29 - 0.347i)T + (1.73 + i)T^{2} \) |
| 7 | \( 1 + (-0.530 + 1.97i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.762 + 0.440i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.43 + 5.36i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-1.13 + 1.13i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.52iT - 19T^{2} \) |
| 23 | \( 1 + (-1.53 + 0.410i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.796 - 1.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.49 + 6.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.25 - 4.25i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.11 + 1.79i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.85 - 0.497i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (7.99 + 2.14i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (4.65 + 4.65i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.81 + 6.61i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.64 - 11.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.20 - 0.859i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 5.89iT - 71T^{2} \) |
| 73 | \( 1 + (1.58 - 1.58i)T - 73iT^{2} \) |
| 79 | \( 1 + (-6.69 + 3.86i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.57 - 9.59i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 + (-1.02 + 3.82i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.20325024634480630258098813171, −9.727827978283105836223602711550, −8.456758373270535527619241115317, −7.59997909314063327879699495801, −6.60463226382779187226302835629, −5.64484782145559452746480464750, −4.86683600724258972937716798191, −3.89482381094659533412115712856, −2.91986866648403568815601863827, −0.794356193456518293637814740135,
1.98820891389972748403330202287, 3.11035580732012498309505181629, 4.30074059047910133507957064828, 4.96929542110802702583012111594, 5.99911670344213182931694278853, 6.92233630838922811938785849252, 8.200202330159966279843950745979, 8.990227477433681575042137031955, 9.647041230734316127889504003107, 10.99476868036295762602966918175