L(s) = 1 | + (1.36 + 0.361i)2-s + (1.26 + 1.26i)3-s + (1.73 + 0.988i)4-s + (1.27 + 2.18i)6-s + (−0.707 + 0.707i)7-s + (2.02 + 1.97i)8-s + 0.204i·9-s − 2.11i·11-s + (0.950 + 3.45i)12-s + (−2.86 + 2.86i)13-s + (−1.22 + 0.711i)14-s + (2.04 + 3.43i)16-s + (4.47 + 4.47i)17-s + (−0.0740 + 0.280i)18-s + 2.95·19-s + ⋯ |
L(s) = 1 | + (0.966 + 0.255i)2-s + (0.730 + 0.730i)3-s + (0.869 + 0.494i)4-s + (0.519 + 0.893i)6-s + (−0.267 + 0.267i)7-s + (0.714 + 0.699i)8-s + 0.0682i·9-s − 0.637i·11-s + (0.274 + 0.996i)12-s + (−0.794 + 0.794i)13-s + (−0.326 + 0.190i)14-s + (0.511 + 0.859i)16-s + (1.08 + 1.08i)17-s + (−0.0174 + 0.0660i)18-s + 0.677·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.68494 + 2.01145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.68494 + 2.01145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.361i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-1.26 - 1.26i)T + 3iT^{2} \) |
| 11 | \( 1 + 2.11iT - 11T^{2} \) |
| 13 | \( 1 + (2.86 - 2.86i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.47 - 4.47i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.95T + 19T^{2} \) |
| 23 | \( 1 + (3.94 + 3.94i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.86iT - 29T^{2} \) |
| 31 | \( 1 + 6.01iT - 31T^{2} \) |
| 37 | \( 1 + (3.91 + 3.91i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 + (5.64 + 5.64i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.30 + 4.30i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.561 + 0.561i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 + (-8.06 + 8.06i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.610iT - 71T^{2} \) |
| 73 | \( 1 + (-3.23 + 3.23i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + (5.51 + 5.51i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.07iT - 89T^{2} \) |
| 97 | \( 1 + (-1.95 - 1.95i)T + 97iT^{2} \) |
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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
−10.49843370501370759350674379425, −9.865401600955517785286933373813, −8.791116045044920932410197093402, −8.083001401498175903399831037569, −6.97747135964045610315077803651, −6.02847169242369158907176573546, −5.10805300863236571899125499471, −3.93547512547416849090670850574, −3.38165868725220388570033189531, −2.17611822741704575098929678753,
1.41661323105861444998198100336, 2.68169705482961991415917775253, 3.39526371863726875963183650119, 4.84148966232142673383260979193, 5.56864941021598566203484209052, 6.99685202888914820143768968252, 7.37777225190129368857698766180, 8.241897278029620847784655073497, 9.877625918760919174776532370565, 10.01885315564016236398830109585