L(s) = 1 | + (−0.707 + 0.707i)3-s + (−1.03 − 1.98i)5-s + (−2.57 + 0.614i)7-s − 1.00i·9-s + 3.85·11-s + (−3.66 + 3.66i)13-s + (2.13 + 0.668i)15-s + (1.49 + 1.49i)17-s − 0.0697·19-s + (1.38 − 2.25i)21-s + (0.534 + 0.534i)23-s + (−2.85 + 4.10i)25-s + (0.707 + 0.707i)27-s − 2.77i·29-s − 2.39i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.463 − 0.886i)5-s + (−0.972 + 0.232i)7-s − 0.333i·9-s + 1.16·11-s + (−1.01 + 1.01i)13-s + (0.550 + 0.172i)15-s + (0.361 + 0.361i)17-s − 0.0160·19-s + (0.302 − 0.491i)21-s + (0.111 + 0.111i)23-s + (−0.570 + 0.821i)25-s + (0.136 + 0.136i)27-s − 0.514i·29-s − 0.430i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.011275805\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011275805\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.03 + 1.98i)T \) |
| 7 | \( 1 + (2.57 - 0.614i)T \) |
good | 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 + (3.66 - 3.66i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.49 - 1.49i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.0697T + 19T^{2} \) |
| 23 | \( 1 + (-0.534 - 0.534i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.77iT - 29T^{2} \) |
| 31 | \( 1 + 2.39iT - 31T^{2} \) |
| 37 | \( 1 + (-6.18 + 6.18i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.68iT - 41T^{2} \) |
| 43 | \( 1 + (-2.77 - 2.77i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.49 + 5.49i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.13 - 6.13i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.97T + 59T^{2} \) |
| 61 | \( 1 + 14.3iT - 61T^{2} \) |
| 67 | \( 1 + (0.416 - 0.416i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + (-9.55 + 9.55i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.86iT - 79T^{2} \) |
| 83 | \( 1 + (1.63 - 1.63i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.05T + 89T^{2} \) |
| 97 | \( 1 + (-6.85 - 6.85i)T + 97iT^{2} \) |
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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
−9.465251079214858623309470965730, −8.740898683387521618506820359303, −7.64475890371555269198761699718, −6.78428116076732834517532960397, −6.02780044166218980423601329179, −5.11979560353563701026058248696, −4.16883549462545929230025564552, −3.64981016566351733031749790162, −2.10667435463428702124867969665, −0.56144630878097495566190186065,
0.884317354470044676764357256191, 2.62678110352628215550065515567, 3.33205575861112184024982206769, 4.36565931053256229872404282370, 5.52322629407407085970496062667, 6.44890311128371903340927844494, 6.94586168108628043209297936331, 7.61214381155761479876355324416, 8.526807137417916875286622156523, 9.819197945497410059979715145625