| L(s) = 1 | + 141. i·3-s − 510·5-s − 2.55e3i·7-s − 1.35e4·9-s + 1.91e4i·11-s − 2.77e4·13-s − 7.24e4i·15-s + 5.03e4·17-s + 1.08e5i·19-s + 3.62e5·21-s + 1.76e5i·23-s − 1.30e5·25-s − 9.99e5i·27-s + 5.49e4·29-s + 1.17e6i·31-s + ⋯ |
| L(s) = 1 | + 1.75i·3-s − 0.816·5-s − 1.06i·7-s − 2.07·9-s + 1.30i·11-s − 0.970·13-s − 1.43i·15-s + 0.603·17-s + 0.833i·19-s + 1.86·21-s + 0.630i·23-s − 0.334·25-s − 1.88i·27-s + 0.0777·29-s + 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.800545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.800545i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 141. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 510T + 3.90e5T^{2} \) |
| 7 | \( 1 + 2.55e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.91e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.77e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.03e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.08e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.76e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 5.49e4T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.17e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 7.93e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 7.55e4T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.99e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 2.86e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.11e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 2.18e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.38e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 7.49e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.00e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 6.51e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.87e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 7.34e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.67e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 4.66e7T + 7.83e15T^{2} \) |
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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
−17.38624172233306483745881687732, −16.40495392749809605101946713050, −15.30052339723711556560907967474, −14.35818415366575748952044509475, −12.05418020255978570190492316653, −10.51838123367464733612617225198, −9.649148753062120836408863498092, −7.62006934548952151741623958571, −4.81358410003335607058665818201, −3.70734191623645052316818471976,
0.43933865474779621515255564421, 2.60379807457954539457520050565, 5.88899677477203516236891189083, 7.49769515905505937595628963416, 8.645736225117550084400133122170, 11.53329504294888990292621947213, 12.27801924036190834159341145690, 13.58529294497711327043524366998, 15.04648351514145639173738149473, 16.76646688473833333705190776647
