L(s) = 1 | − 9.43i·2-s + (80.9 + 1.77i)3-s + 167.·4-s − 654. i·5-s + (16.6 − 763. i)6-s − 1.00e3·7-s − 3.99e3i·8-s + (6.55e3 + 286. i)9-s − 6.17e3·10-s + 4.41e3i·11-s + (1.35e4 + 295. i)12-s + 1.56e4·13-s + 9.47e3i·14-s + (1.15e3 − 5.29e4i)15-s + 5.10e3·16-s − 2.86e4i·17-s + ⋯ |
L(s) = 1 | − 0.589i·2-s + (0.999 + 0.0218i)3-s + 0.652·4-s − 1.04i·5-s + (0.0128 − 0.589i)6-s − 0.418·7-s − 0.974i·8-s + (0.999 + 0.0436i)9-s − 0.617·10-s + 0.301i·11-s + (0.652 + 0.0142i)12-s + 0.549·13-s + 0.246i·14-s + (0.0228 − 1.04i)15-s + 0.0779·16-s − 0.343i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0218 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0218 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.03484 - 2.07981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03484 - 2.07981i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-80.9 - 1.77i)T \) |
| 11 | \( 1 - 4.41e3iT \) |
good | 2 | \( 1 + 9.43iT - 256T^{2} \) |
| 5 | \( 1 + 654. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.00e3T + 5.76e6T^{2} \) |
| 13 | \( 1 - 1.56e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 2.86e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.55e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.98e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 8.41e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.36e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.28e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.48e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.46e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 4.25e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.30e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 2.01e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.44e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.96e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.15e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.15e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 1.18e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 8.83e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 6.59e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 9.98e7T + 7.83e15T^{2} \) |
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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
−14.62518566176833343366854155467, −12.98930714901158305373594998239, −12.50317987903073102917139530277, −10.75324956656815289363727123213, −9.462528301692006582747218212755, −8.303471723296784407920876117534, −6.65856172767297937537304897071, −4.30731271723911924273991018513, −2.69294700521151377473858606468, −1.19558602180601278704790700304,
2.18230832893404733471237735502, 3.52073590197196507915072455834, 6.18516528351006095699605947749, 7.23722945002991523377739526378, 8.455143376718772382136313150210, 10.13899350811322942450396590077, 11.34370535874242238277469814317, 13.15207872941440816497763048531, 14.35976113920690821277669868441, 15.18784766845840859709728942184