| L(s) = 1 | + 878. i·5-s + 1.52e3·7-s − 1.01e4i·11-s − 7.95e3·13-s + 9.94e4i·17-s + 1.05e5·19-s − 2.04e5i·23-s − 3.81e5·25-s + 4.32e5i·29-s + 1.06e6·31-s + 1.33e6i·35-s + 3.38e6·37-s + 2.50e6i·41-s + 6.56e6·43-s − 8.43e6i·47-s + ⋯ |
| L(s) = 1 | + 1.40i·5-s + 0.634·7-s − 0.694i·11-s − 0.278·13-s + 1.19i·17-s + 0.807·19-s − 0.731i·23-s − 0.977·25-s + 0.612i·29-s + 1.15·31-s + 0.892i·35-s + 1.80·37-s + 0.887i·41-s + 1.92·43-s − 1.72i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.434523521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434523521\) |
| \(L(5)\) |
|
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 | \( 1 - 878. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.52e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.01e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 7.95e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 9.94e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.05e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.04e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 4.32e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.06e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.38e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.50e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.56e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 8.43e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.67e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 3.81e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.51e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.27e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.56e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.28e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.26e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 7.07e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 6.81e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 4.47e7T + 7.83e15T^{2} \) |
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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
−10.65534309921497161171784879842, −9.706165799833141813488334657672, −8.411824290668742360516826973017, −7.63625769975655039198025540288, −6.59324146480464332114987833381, −5.80499231712798552535941435758, −4.43238894150370740779154393448, −3.25437230799707099764719368514, −2.38615420900548481163631898704, −1.00589612474288128278077552229,
0.58400016901776535189870221275, 1.40152188287431314168932426022, 2.67996097843447895802829838776, 4.38069689531042784988885059820, 4.87285599448851524929704108528, 5.89187391759015083808221601973, 7.44258074324749514839928157660, 8.009219528407905226953108827636, 9.299833629268881993984740009027, 9.588799337475076191950594882375
