L(s) = 1 | + (0.707 − 0.707i)3-s + (1.03 + 1.98i)5-s + (−0.614 + 2.57i)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 + (−0.232 + 0.972i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.350041426\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.350041426\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.614 - 2.57i)T \) |
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
−9.266895449303878172551440031315, −8.770516941004815860656211431341, −7.80369460536198924727885141570, −6.97649917522793822628385529120, −6.03251374619028176922211445988, −5.84175672121484702931292188624, −4.24266366447702310782601921755, −3.16139928007197552948113100464, −2.54507207899540097527357870546, −1.29168434142689295102366881641,
1.02205076281411068463562019929, 2.02541680437625225546900250374, 3.71288772739513424561379935963, 4.07641030639433167364953672435, 5.00166549665087971078513031043, 6.21001291352217136838431136526, 6.74437069683709161568425492373, 7.85559335627840692992457889399, 8.810274442378678286435846144080, 9.132492428835346627449042827129