L(s) = 1 | + (−240. − 345. i)3-s + 113.·5-s + (4.40e4 + 6.36e3i)7-s + (−6.15e4 + 1.66e5i)9-s − 4.71e5i·11-s + 1.52e6i·13-s + (−2.72e4 − 3.91e4i)15-s − 1.63e6·17-s + 2.97e6i·19-s + (−8.38e6 − 1.67e7i)21-s − 2.79e7i·23-s − 4.88e7·25-s + (7.21e7 − 1.86e7i)27-s − 1.72e8i·29-s + 6.77e7i·31-s + ⋯ |
L(s) = 1 | + (−0.571 − 0.820i)3-s + 0.0162·5-s + (0.989 + 0.143i)7-s + (−0.347 + 0.937i)9-s − 0.883i·11-s + 1.13i·13-s + (−0.00926 − 0.0133i)15-s − 0.278·17-s + 0.275i·19-s + (−0.447 − 0.894i)21-s − 0.904i·23-s − 0.999·25-s + (0.968 − 0.250i)27-s − 1.56i·29-s + 0.424i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.193224 + 0.312904i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.193224 + 0.312904i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (240. + 345. i)T \) |
| 7 | \( 1 + (-4.40e4 - 6.36e3i)T \) |
good | 5 | \( 1 - 113.T + 4.88e7T^{2} \) |
| 11 | \( 1 + 4.71e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 - 1.52e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 1.63e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.97e6iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 2.79e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 1.72e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 - 6.77e7iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 2.11e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.13e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.22e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.73e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.10e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 9.33e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.27e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 - 9.31e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.13e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 1.18e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 2.57e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.83e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.88e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.23e11iT - 7.15e21T^{2} \) |
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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
−12.07076369304741725108699482576, −11.54047828920768965736027625308, −10.50178787783779064835907703438, −8.785089444884569233599622194671, −7.87697186588248409212715997792, −6.62724170185958351791679371229, −5.58502785884973588147351148387, −4.30084572185501756214956749368, −2.32596928508157933596360628448, −1.28488226791351302451638756847,
0.098718651309257569987787873372, 1.65194321279969916464374737557, 3.44677301391012612446080448778, 4.74951192033669711572035989238, 5.52515695976645119675718048879, 7.10702087210415341439539516012, 8.385739046471437624700564151956, 9.691284954166882032204948169228, 10.60752178795176897270811471619, 11.51411071978497601212269676285