| L(s) = 1 | + 671. i·5-s + 3.82e3·7-s − 2.90e3i·11-s + 1.66e4·13-s + 7.97e4i·17-s + 1.98e5·19-s − 5.53e5i·23-s − 6.08e4·25-s − 2.04e5i·29-s + 1.19e6·31-s + 2.57e6i·35-s + 3.26e5·37-s − 2.29e6i·41-s + 1.93e6·43-s − 6.14e6i·47-s + ⋯ |
| L(s) = 1 | + 1.07i·5-s + 1.59·7-s − 0.198i·11-s + 0.584·13-s + 0.954i·17-s + 1.52·19-s − 1.97i·23-s − 0.155·25-s − 0.288i·29-s + 1.28·31-s + 1.71i·35-s + 0.174·37-s − 0.812i·41-s + 0.565·43-s − 1.25i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.435073840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.435073840\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 671. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 3.82e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.90e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.66e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 7.97e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.98e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 5.53e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 2.04e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.19e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.26e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.29e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.93e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.14e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.38e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.15e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.36e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.33e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 7.52e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.32e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.06e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.33e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.02e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 3.00e7T + 7.83e15T^{2} \) |
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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
−10.10756543665298449118549523194, −8.634189707813729155360997703296, −8.084474881331656477653740407585, −7.07587804372641840815607785726, −6.15187698075946603670553081322, −5.07270850256641604162918309965, −4.04310708378972908799500256155, −2.88346498717756419953465572281, −1.84893780742691448368183324519, −0.73891912677531575771037053032,
1.17520367771101033795350199835, 1.27340998121651347104172286911, 2.91574612899314976555252970988, 4.35612942977447757759033551375, 5.02182409748415014231688053883, 5.76332274849155360854420234186, 7.43146760488888841396105064348, 7.914718451976931334793526301162, 8.956606663750312505419981954405, 9.569134551936556276285333042306
