| L(s) = 1 | − 32i·2-s − 141. i·3-s − 1.02e3·4-s − 4.53e3·6-s − 8.55e4i·7-s + 3.27e4i·8-s + 1.57e5·9-s + 7.67e5·11-s + 1.45e5i·12-s + 2.20e5i·13-s − 2.73e6·14-s + 1.04e6·16-s + 9.30e5i·17-s − 5.02e6i·18-s + 1.77e7·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.336i·3-s − 0.5·4-s − 0.238·6-s − 1.92i·7-s + 0.353i·8-s + 0.886·9-s + 1.43·11-s + 0.168i·12-s + 0.165i·13-s − 1.36·14-s + 0.250·16-s + 0.158i·17-s − 0.626i·18-s + 1.64·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.506147 - 2.14407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.506147 - 2.14407i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 141. iT - 1.77e5T^{2} \) |
| 7 | \( 1 + 8.55e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 - 7.67e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.20e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 9.30e5iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.77e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.99e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 7.68e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.96e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.40e7iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 1.26e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.88e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 1.57e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 4.09e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 3.77e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.64e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.63e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 1.03e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 4.27e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 1.96e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.35e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 2.25e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.08e11iT - 7.15e21T^{2} \) |
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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
−12.68359625854004849864126778390, −11.46936216035145096291583878827, −10.36050374896010381237996729370, −9.412286448099405512169171674916, −7.62717640895437629849894616454, −6.68739769356415000333783625127, −4.46322747310214952894127242970, −3.61169204392278969436927999702, −1.48337359815727072701304889333, −0.73979535553064904710692539300,
1.51668681609890157242233417884, 3.43710618960350994203144341868, 5.05347098119580774683187870377, 6.09389380125278572618151989466, 7.51696931984512297197563233149, 9.112281229427588618138833445905, 9.530438071092830605519095711025, 11.57636863911212645061299825601, 12.43619589698574426378025521426, 13.87809222820084793150289553547
