L(s) = 1 | + 1.02e5i·3-s + 2.09e6·4-s + (−5.61e8 + 4.92e8i)7-s − 1.04e10·9-s + 2.14e11i·12-s − 9.22e11i·13-s + 4.39e12·16-s − 3.99e13i·19-s + (−5.03e13 − 5.74e13i)21-s − 4.76e14·25-s − 1.06e15i·27-s + (−1.17e15 + 1.03e15i)28-s + 1.25e15i·31-s − 2.19e16·36-s + 5.77e16·37-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + 4-s + (−0.751 + 0.659i)7-s − 0.999·9-s + 0.999i·12-s − 1.85i·13-s + 16-s − 1.49i·19-s + (−0.659 − 0.751i)21-s − 0.999·25-s − 0.999i·27-s + (−0.751 + 0.659i)28-s + 0.275i·31-s − 0.999·36-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(1.587701918\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.587701918\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.02e5iT \) |
| 7 | \( 1 + (5.61e8 - 4.92e8i)T \) |
good | 2 | \( 1 - 2.09e6T^{2} \) |
| 5 | \( 1 + 4.76e14T^{2} \) |
| 11 | \( 1 - 7.40e21T^{2} \) |
| 13 | \( 1 + 9.22e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.99e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 3.94e28T^{2} \) |
| 29 | \( 1 - 5.13e30T^{2} \) |
| 31 | \( 1 - 1.25e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 5.77e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.65e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.62e36T^{2} \) |
| 59 | \( 1 + 1.54e37T^{2} \) |
| 61 | \( 1 + 2.49e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 - 6.94e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 7.52e38T^{2} \) |
| 73 | \( 1 + 6.21e19iT - 1.34e39T^{2} \) |
| 79 | \( 1 - 1.68e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.99e40T^{2} \) |
| 89 | \( 1 + 8.65e40T^{2} \) |
| 97 | \( 1 + 9.10e20iT - 5.27e41T^{2} \) |
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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
−13.09818957488653745666804662523, −11.72013349629435612846346563264, −10.60840162707304358748483100720, −9.548860987763123891895986677996, −8.028631361168046090140950915120, −6.30491965200978824264409596211, −5.24498372983927034602958941826, −3.32841598919870425379346546950, −2.54615738994233115887787312201, −0.38331335069110227525029462216,
1.25900613417304301123181429083, 2.24979295557101562801861921486, 3.74850258106285630385815220731, 6.07978921998074230045655644739, 6.83550984800372671030370508226, 7.928099016901005355392369570320, 9.753470761907467476116248018365, 11.31691848112805478908525738983, 12.19200834271026321305495542484, 13.49569354382904190930906419692