L(s) = 1 | − 1.09·2-s + (−0.323 + 1.70i)3-s − 0.791·4-s + (0.355 − 1.87i)6-s + (−2.44 + i)7-s + 3.06·8-s + (−2.79 − 1.09i)9-s − 3.06i·11-s + (0.255 − 1.34i)12-s − 2.44·13-s + (2.69 − 1.09i)14-s − 1.79·16-s + 2.69i·17-s + (3.06 + 1.20i)18-s − 4.38i·19-s + ⋯ |
L(s) = 1 | − 0.777·2-s + (−0.186 + 0.982i)3-s − 0.395·4-s + (0.144 − 0.763i)6-s + (−0.925 + 0.377i)7-s + 1.08·8-s + (−0.930 − 0.366i)9-s − 0.925i·11-s + (0.0737 − 0.388i)12-s − 0.679·13-s + (0.719 − 0.293i)14-s − 0.447·16-s + 0.653i·17-s + (0.723 + 0.284i)18-s − 1.00i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.351514 - 0.171355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.351514 - 0.171355i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.323 - 1.70i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.44 - i)T \) |
good | 2 | \( 1 + 1.09T + 2T^{2} \) |
| 11 | \( 1 + 3.06iT - 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 2.69iT - 17T^{2} \) |
| 19 | \( 1 + 4.38iT - 19T^{2} \) |
| 23 | \( 1 - 5.26T + 23T^{2} \) |
| 29 | \( 1 + 5.26iT - 29T^{2} \) |
| 31 | \( 1 - 6.83iT - 31T^{2} \) |
| 37 | \( 1 + 8.58iT - 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 6.58iT - 43T^{2} \) |
| 47 | \( 1 - 2.69iT - 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 - 7.51T + 59T^{2} \) |
| 61 | \( 1 + 6.83iT - 61T^{2} \) |
| 67 | \( 1 + 4.16iT - 67T^{2} \) |
| 71 | \( 1 + 3.06iT - 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 0.582T + 79T^{2} \) |
| 83 | \( 1 + 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 7.51T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.58892511433837331127447514136, −9.680365367147459794865701092279, −9.079680979221772121013474479724, −8.524730046909973978510471263811, −7.23382801866274510408474722097, −6.00745431916938627205591387578, −5.06691589523625262163721011026, −3.97380758937012775883184078192, −2.82224520617536495879128557205, −0.35665445380557668384592373830,
1.18907100932464992625011913349, 2.75243534002752259158826003568, 4.36040461449478338920404724079, 5.52464623509828911640249964853, 6.84223217521093326842151945525, 7.36511434438801501060380747485, 8.221619472599239599471173391526, 9.351465913099474924436199169747, 9.864383688255536601531366199487, 10.82619397427794293468840901614