L(s) = 1 | − 2.32e3i·2-s + (−5.29e4 + 8.75e4i)3-s − 3.33e6·4-s + 3.45e7·5-s + (2.03e8 + 1.23e8i)6-s + (4.88e7 + 7.45e8i)7-s + 2.87e9i·8-s + (−4.86e9 − 9.26e9i)9-s − 8.03e10i·10-s + 5.12e9i·11-s + (1.76e11 − 2.91e11i)12-s − 3.88e11i·13-s + (1.73e12 − 1.13e11i)14-s + (−1.82e12 + 3.02e12i)15-s − 2.90e11·16-s + 4.11e12·17-s + ⋯ |
L(s) = 1 | − 1.60i·2-s + (−0.517 + 0.855i)3-s − 1.58·4-s + 1.58·5-s + (1.37 + 0.832i)6-s + (0.0653 + 0.997i)7-s + 0.946i·8-s + (−0.464 − 0.885i)9-s − 2.54i·10-s + 0.0596i·11-s + (0.821 − 1.35i)12-s − 0.781i·13-s + (1.60 − 0.105i)14-s + (−0.817 + 1.35i)15-s − 0.0660·16-s + 0.494·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.075836727\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.075836727\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.29e4 - 8.75e4i)T \) |
| 7 | \( 1 + (-4.88e7 - 7.45e8i)T \) |
good | 2 | \( 1 + 2.32e3iT - 2.09e6T^{2} \) |
| 5 | \( 1 - 3.45e7T + 4.76e14T^{2} \) |
| 11 | \( 1 - 5.12e9iT - 7.40e21T^{2} \) |
| 13 | \( 1 + 3.88e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 - 4.11e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 7.35e11iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 2.80e14iT - 3.94e28T^{2} \) |
| 29 | \( 1 + 2.85e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 8.55e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 + 2.16e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 8.87e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.16e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.24e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.11e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 3.06e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.59e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 - 9.86e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.59e19iT - 7.52e38T^{2} \) |
| 73 | \( 1 - 2.69e18iT - 1.34e39T^{2} \) |
| 79 | \( 1 - 5.09e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.71e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.53e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.89e20iT - 5.27e41T^{2} \) |
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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
−12.89120317082331394625009558920, −11.85173386328424903961985564561, −10.60546925287147249240727452029, −9.779670121936746990324635737887, −9.016522255487274767971819750475, −5.92070309899977060257146693903, −5.06570420097288503518211201171, −3.32221792686622903898033419929, −2.26450937632990198874256584181, −1.03997537408073650373626009411,
0.66107078623176051936865343689, 2.04932603741552353686585995638, 4.74177749812631696598427789044, 5.94648142273437193014052558002, 6.63845816784189460203155296334, 7.71501360163057335953127588741, 9.233149718898815681751263745298, 10.74527641623290502217687424948, 12.82567604345639861438554545363, 13.90172540228635286315900076075