L(s) = 1 | + (1.28e3 − 672. i)2-s − 6.03e4i·3-s + (1.19e6 − 1.72e6i)4-s + 1.19e7i·5-s + (−4.05e7 − 7.73e7i)6-s + 8.57e8·7-s + (3.69e8 − 3.01e9i)8-s + 6.82e9·9-s + (8.03e9 + 1.53e10i)10-s − 3.73e10i·11-s + (−1.04e11 − 7.19e10i)12-s − 4.68e11i·13-s + (1.09e12 − 5.76e11i)14-s + 7.20e11·15-s + (−1.55e12 − 4.11e12i)16-s − 7.52e12·17-s + ⋯ |
L(s) = 1 | + (0.885 − 0.464i)2-s − 0.589i·3-s + (0.568 − 0.822i)4-s + 0.546i·5-s + (−0.273 − 0.522i)6-s + 1.14·7-s + (0.121 − 0.992i)8-s + 0.652·9-s + (0.253 + 0.484i)10-s − 0.433i·11-s + (−0.485 − 0.335i)12-s − 0.942i·13-s + (1.01 − 0.532i)14-s + 0.322·15-s + (−0.353 − 0.935i)16-s − 0.905·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.50251 - 2.82765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.50251 - 2.82765i\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.28e3 + 672. i)T \) |
good | 3 | \( 1 + 6.03e4iT - 1.04e10T^{2} \) |
| 5 | \( 1 - 1.19e7iT - 4.76e14T^{2} \) |
| 7 | \( 1 - 8.57e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.73e10iT - 7.40e21T^{2} \) |
| 13 | \( 1 + 4.68e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 + 7.52e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.69e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 + 1.71e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.63e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 1.78e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.66e16iT - 8.55e32T^{2} \) |
| 41 | \( 1 - 8.59e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.20e17iT - 2.00e34T^{2} \) |
| 47 | \( 1 - 3.64e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.49e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 2.76e18iT - 1.54e37T^{2} \) |
| 61 | \( 1 - 3.78e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 - 2.47e19iT - 2.22e38T^{2} \) |
| 71 | \( 1 + 1.31e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.62e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.88e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.83e20iT - 1.99e40T^{2} \) |
| 89 | \( 1 + 1.06e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.14e21T + 5.27e41T^{2} \) |
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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
−15.57090182052240924198979665520, −14.32677185972219354429933845442, −13.08525155168354620504927248176, −11.63477575024164599784613924792, −10.37901915809672215493058273435, −7.74979924829918070505849733375, −6.11840820908660081934778720360, −4.35577295356355664745250856808, −2.46708807606223595285953605055, −1.08939655775878604980774193881,
1.87372401437026518150448432406, 4.20102866199878606011852251641, 4.98535187466030257138186777782, 7.08365683266143971966145696619, 8.817601337561381160963562369300, 11.02029788596951980379382139663, 12.49827899833447173233712332770, 14.07138098531536759237958302869, 15.33315461748192171908808092785, 16.46358989873643622792847826415