| L(s) = 1 | + 46.7·3-s − 402. i·5-s − 1.93e3i·7-s + 2.18e3·9-s + 2.01e4·11-s + 4.37e4i·13-s − 1.88e4i·15-s + 1.01e5·17-s + 1.39e4·19-s − 9.05e4i·21-s − 4.33e5i·23-s + 2.28e5·25-s + 1.02e5·27-s + 4.67e5i·29-s + 1.48e6i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.643i·5-s − 0.806i·7-s + 0.333·9-s + 1.37·11-s + 1.53i·13-s − 0.371i·15-s + 1.21·17-s + 0.106·19-s − 0.465i·21-s − 1.54i·23-s + 0.585·25-s + 0.192·27-s + 0.661i·29-s + 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.550486315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550486315\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 46.7T \) |
| good | 5 | \( 1 + 402. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.93e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.01e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 4.37e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.01e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.39e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.33e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 4.67e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.48e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.57e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 3.84e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 1.88e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.93e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 8.07e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.78e7T + 1.46e14T^{2} \) |
| 61 | \( 1 - 2.11e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 7.11e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.36e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.97e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 8.29e5iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 1.45e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 7.77e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 9.95e7T + 7.83e15T^{2} \) |
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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
−9.820341358599418595422762578732, −8.978137447633365276374326590469, −8.375641589138967049245891443638, −7.05667490420258340914032995429, −6.52527729557760759916346401017, −4.86600579139335203477175309263, −4.15343507322864744099037989709, −3.16127598827928969080457603607, −1.55551913761489378255644078296, −1.01016221305228105447770612616,
0.74875471503624393377991841243, 1.98117829297000505904515254093, 3.11520600731085162838610410974, 3.75234809083967875536245891303, 5.37021419131121469421175742695, 6.13640561937486855743642760299, 7.36871149634003167839659809914, 8.059572605174730978332987175276, 9.189985884201122558551997364745, 9.791186939464807680070728227594
