L(s) = 1 | − 77.4·3-s + (−586. + 214. i)5-s − 2.76e3·7-s − 560.·9-s − 1.87e4i·11-s − 3.86e4i·13-s + (4.54e4 − 1.66e4i)15-s + 7.64e4i·17-s − 3.35e4i·19-s + 2.14e5·21-s + 2.22e5·23-s + (2.98e5 − 2.52e5i)25-s + 5.51e5·27-s − 6.51e5·29-s + 1.03e6i·31-s + ⋯ |
L(s) = 1 | − 0.956·3-s + (−0.939 + 0.343i)5-s − 1.15·7-s − 0.0853·9-s − 1.28i·11-s − 1.35i·13-s + (0.898 − 0.328i)15-s + 0.915i·17-s − 0.257i·19-s + 1.10·21-s + 0.794·23-s + (0.763 − 0.645i)25-s + 1.03·27-s − 0.921·29-s + 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3600645426\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3600645426\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (586. - 214. i)T \) |
good | 3 | \( 1 + 77.4T + 6.56e3T^{2} \) |
| 7 | \( 1 + 2.76e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.87e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.86e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 7.64e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 3.35e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.22e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 6.51e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.03e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.60e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 4.22e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 8.65e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 1.55e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.13e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.73e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 6.27e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.69e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.42e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.21e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 3.05e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.71e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 3.34e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.14e8iT - 7.83e15T^{2} \) |
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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
−10.14777610035266710229162605259, −8.768415024700374006477626326607, −7.944422374352314058770743663741, −6.68887233794269217416394284164, −6.03311121993588781758155945006, −5.07229245469310694955679533728, −3.49403454733362098446211696149, −3.03497180171594140073788661297, −0.76639919628342030049524992822, −0.16332192153734602557766605620,
0.827352028073636774093556062964, 2.46195571402197625376449583545, 3.89312038409661459839286228411, 4.69966959866147030656722674415, 5.83487099814793452887261103554, 6.90408843093040125008452628660, 7.48563976692328872068197735995, 9.095523398258566079588189340081, 9.570504116805626532566235875734, 10.92539853072614143449222457598