| L(s) = 1 | + 823. i·5-s − 1.96e3·7-s − 1.25e4i·11-s − 4.55e4·13-s + 5.96e4i·17-s − 1.52e5·19-s − 1.31e5i·23-s − 2.86e5·25-s + 5.88e5i·29-s + 1.64e5·31-s − 1.61e6i·35-s − 6.63e5·37-s + 9.38e5i·41-s − 5.75e5·43-s − 9.23e6i·47-s + ⋯ |
| L(s) = 1 | + 1.31i·5-s − 0.819·7-s − 0.859i·11-s − 1.59·13-s + 0.713i·17-s − 1.16·19-s − 0.469i·23-s − 0.734·25-s + 0.832i·29-s + 0.177·31-s − 1.07i·35-s − 0.354·37-s + 0.331i·41-s − 0.168·43-s − 1.89i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8048344831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8048344831\) |
| \(L(5)\) |
|
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 | \( 1 - 823. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.96e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.25e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.55e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.96e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.52e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.31e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 5.88e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.64e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 6.63e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 9.38e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.75e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 9.23e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.03e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 5.03e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.92e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 5.98e5T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.92e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.28e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.35e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.34e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 2.82e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.36e8T + 7.83e15T^{2} \) |
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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
−10.11707696218553024557046599395, −8.953949012315052238195958896037, −7.914348943283619604862540139932, −6.76998243747673147144222584874, −6.44982829422425594076221132908, −5.16938595547017311873622864617, −3.77800815465976087221927276695, −2.94998090021063049114255439890, −2.10123225487358486837748615374, −0.28622529491760750813658653149,
0.46703823096191349754737988780, 1.78948603494813160069251165504, 2.84028663720564062019123858595, 4.38297465733266635445616794391, 4.85568679679269071266729296969, 6.03241062742528007166365358209, 7.15005147571311955228468794864, 7.966897557876922784127134498676, 9.237388879609421002545183655210, 9.514544384008696011116036202269
