L(s) = 1 | + (−2.05e6 − 4.25e5i)2-s + 8.66e9i·3-s + (4.03e12 + 1.74e12i)4-s − 5.87e14·5-s + (3.68e15 − 1.77e16i)6-s − 1.08e18i·7-s + (−7.54e18 − 5.30e18i)8-s + 3.43e19·9-s + (1.20e21 + 2.49e20i)10-s − 1.54e21i·11-s + (−1.51e22 + 3.49e22i)12-s − 2.11e23·13-s + (−4.60e23 + 2.22e24i)14-s − 5.09e24i·15-s + (1.32e25 + 1.40e25i)16-s + 9.80e25·17-s + ⋯ |
L(s) = 1 | + (−0.979 − 0.202i)2-s + 0.828i·3-s + (0.917 + 0.397i)4-s − 1.23·5-s + (0.167 − 0.811i)6-s − 1.94i·7-s + (−0.818 − 0.574i)8-s + 0.313·9-s + (1.20 + 0.249i)10-s − 0.209i·11-s + (−0.328 + 0.760i)12-s − 0.857·13-s + (−0.393 + 1.90i)14-s − 1.02i·15-s + (0.684 + 0.728i)16-s + 1.41·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{43}{2})\) |
\(\approx\) |
\(0.002795545580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002795545580\) |
\(L(22)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.05e6 + 4.25e5i)T \) |
good | 3 | \( 1 - 8.66e9iT - 1.09e20T^{2} \) |
| 5 | \( 1 + 5.87e14T + 2.27e29T^{2} \) |
| 7 | \( 1 + 1.08e18iT - 3.11e35T^{2} \) |
| 11 | \( 1 + 1.54e21iT - 5.47e43T^{2} \) |
| 13 | \( 1 + 2.11e23T + 6.10e46T^{2} \) |
| 17 | \( 1 - 9.80e25T + 4.77e51T^{2} \) |
| 19 | \( 1 - 6.85e26iT - 5.10e53T^{2} \) |
| 23 | \( 1 + 7.24e28iT - 1.55e57T^{2} \) |
| 29 | \( 1 - 5.49e29T + 2.63e61T^{2} \) |
| 31 | \( 1 + 3.73e30iT - 4.33e62T^{2} \) |
| 37 | \( 1 + 5.40e32T + 7.31e65T^{2} \) |
| 41 | \( 1 - 6.20e32T + 5.45e67T^{2} \) |
| 43 | \( 1 - 1.07e33iT - 4.03e68T^{2} \) |
| 47 | \( 1 + 2.86e34iT - 1.69e70T^{2} \) |
| 53 | \( 1 - 1.08e36T + 2.62e72T^{2} \) |
| 59 | \( 1 - 7.58e36iT - 2.37e74T^{2} \) |
| 61 | \( 1 + 2.45e37T + 9.63e74T^{2} \) |
| 67 | \( 1 + 2.23e38iT - 4.95e76T^{2} \) |
| 71 | \( 1 - 4.36e38iT - 5.66e77T^{2} \) |
| 73 | \( 1 + 1.36e39T + 1.81e78T^{2} \) |
| 79 | \( 1 - 5.82e38iT - 5.01e79T^{2} \) |
| 83 | \( 1 + 6.65e39iT - 3.99e80T^{2} \) |
| 89 | \( 1 + 3.51e40T + 7.48e81T^{2} \) |
| 97 | \( 1 + 6.45e41T + 2.78e83T^{2} \) |
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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
−14.67527138596664400108356621777, −12.26158904844139131993645616892, −10.66948658598352264074176838503, −10.00674035166926400066702076595, −7.990467908329985157050451038919, −7.12407176774310141519099709601, −4.29713062374977908701930073495, −3.43249526437283645842338802487, −1.04071465763882415782031207998, −0.00130271441262928733071060248,
1.54994473268172913790997425568, 2.86352451578898157777156145118, 5.47930329791952300019854995912, 7.16441747453916469333809649407, 8.087077667091141318383073914455, 9.490428184744792080846382350594, 11.72173764430285417704850032900, 12.28158444054533487774203119650, 15.06426400361228417000359221584, 15.83224460396407859142905421598