L(s) = 1 | + 2.06e3i·2-s + (−759. + 1.02e5i)3-s − 2.15e6·4-s + 1.95e7·5-s + (−2.10e8 − 1.56e6i)6-s + (−6.67e8 + 3.36e8i)7-s − 1.15e8i·8-s + (−1.04e10 − 1.55e8i)9-s + 4.02e10i·10-s + 1.09e11i·11-s + (1.63e9 − 2.20e11i)12-s + 5.15e11i·13-s + (−6.93e11 − 1.37e12i)14-s + (−1.48e10 + 1.99e12i)15-s − 4.27e12·16-s + 1.12e13·17-s + ⋯ |
L(s) = 1 | + 1.42i·2-s + (−0.00742 + 0.999i)3-s − 1.02·4-s + 0.894·5-s + (−1.42 − 0.0105i)6-s + (−0.892 + 0.450i)7-s − 0.0381i·8-s + (−0.999 − 0.0148i)9-s + 1.27i·10-s + 1.27i·11-s + (0.00762 − 1.02i)12-s + 1.03i·13-s + (−0.640 − 1.27i)14-s + (−0.00663 + 0.893i)15-s − 0.972·16-s + 1.35·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(1.754229962\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.754229962\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (759. - 1.02e5i)T \) |
| 7 | \( 1 + (6.67e8 - 3.36e8i)T \) |
good | 2 | \( 1 - 2.06e3iT - 2.09e6T^{2} \) |
| 5 | \( 1 - 1.95e7T + 4.76e14T^{2} \) |
| 11 | \( 1 - 1.09e11iT - 7.40e21T^{2} \) |
| 13 | \( 1 - 5.15e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 - 1.12e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.21e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 2.30e14iT - 3.94e28T^{2} \) |
| 29 | \( 1 + 2.95e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 + 4.09e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 2.53e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.05e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.13e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.13e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.61e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 - 3.38e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.43e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 - 1.42e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.98e19iT - 7.52e38T^{2} \) |
| 73 | \( 1 - 3.08e18iT - 1.34e39T^{2} \) |
| 79 | \( 1 + 6.07e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 3.37e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.45e18T + 8.65e40T^{2} \) |
| 97 | \( 1 - 8.29e20iT - 5.27e41T^{2} \) |
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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
−14.72851980643439079453525066866, −13.76931277671634684580232615974, −11.98396294999280061681090317780, −9.842180050116080682961778250559, −9.439820670631505574461518738203, −7.74166643611869804041391394927, −6.16421218824321433588411590784, −5.51684993638844411760417497840, −4.01509820344817169677265847787, −2.16362753435409455250603308067,
0.49819320365068265658631537738, 1.05449158646982278957496208484, 2.61752930597003821679324203879, 3.26479457908998657137648565701, 5.60712172818302279894825006772, 6.87536791486537910207710818139, 8.726541611231350734710211657762, 10.08647135021913605317270692499, 11.07160843907525823895708631399, 12.51572015310278563058876589484