L(s) = 1 | + (0.361 + 1.36i)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 + (−1.97 − 2.02i)8-s + 0.204i·9-s + 2.11i·11-s + (3.45 + 0.950i)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 + (−0.280 + 0.0740i)18-s − 2.95·19-s + ⋯ |
L(s) = 1 | + (0.255 + 0.966i)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 + (−0.699 − 0.714i)8-s + 0.0682i·9-s + 0.637i·11-s + (0.996 + 0.274i)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 + (−0.0660 + 0.0174i)18-s − 0.677·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332717 + 0.763138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332717 + 0.763138i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.361 - 1.36i)T \) |
| 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
−10.80893864885405293072818558821, −9.761562225601084989284685733545, −8.889350768128884848329821387801, −7.79710204649220387239401008470, −7.10678491689136485272724181658, −6.49615133705790005911172130762, −5.50740674114857734868975927629, −4.67928359501865441957568249650, −3.50440361593799523334081384353, −1.48495230487374791703257368352,
0.46463494162143809893765694369, 2.39160740750634140493926343391, 3.47780320384936432436205194856, 4.78127150502046754995450867784, 5.20291648851257963403209026726, 6.12518394952558492991469722916, 7.69901625945361384008496736373, 8.662111965535182752519191655183, 9.670899597866540227241935829978, 10.31110031544291262726343400701