L(s) = 1 | + 5.13i·3-s + (−2.79 − 4.14i)5-s + 6.19·7-s − 17.3·9-s − 20.0·11-s − 15.8·13-s + (21.2 − 14.3i)15-s − 6.98i·17-s + 10.3·19-s + 31.7i·21-s − 22.3·23-s + (−9.35 + 23.1i)25-s − 42.9i·27-s − 4.20i·29-s − 20.7i·31-s + ⋯ |
L(s) = 1 | + 1.71i·3-s + (−0.559 − 0.828i)5-s + 0.884·7-s − 1.92·9-s − 1.82·11-s − 1.22·13-s + (1.41 − 0.957i)15-s − 0.411i·17-s + 0.546·19-s + 1.51i·21-s − 0.973·23-s + (−0.374 + 0.927i)25-s − 1.58i·27-s − 0.145i·29-s − 0.668i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.754i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0577230 - 0.126635i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0577230 - 0.126635i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.79 + 4.14i)T \) |
good | 3 | \( 1 - 5.13iT - 9T^{2} \) |
| 7 | \( 1 - 6.19T + 49T^{2} \) |
| 11 | \( 1 + 20.0T + 121T^{2} \) |
| 13 | \( 1 + 15.8T + 169T^{2} \) |
| 17 | \( 1 + 6.98iT - 289T^{2} \) |
| 19 | \( 1 - 10.3T + 361T^{2} \) |
| 23 | \( 1 + 22.3T + 529T^{2} \) |
| 29 | \( 1 + 4.20iT - 841T^{2} \) |
| 31 | \( 1 + 20.7iT - 961T^{2} \) |
| 37 | \( 1 - 35.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 37.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 23.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 48.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 77.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 0.497T + 3.48e3T^{2} \) |
| 61 | \( 1 + 60.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 82.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 28.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 87.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 49.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 84.4iT - 9.40e3T^{2} \) |
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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
−11.71461849433424751830373298119, −11.02467802427131518048057120970, −10.03030297973526689165846382767, −9.453075555972198693528146390379, −8.198922199986965136036451465082, −7.72778428229986686622264021228, −5.40585038921596457800996433826, −4.98116902846443513662703600371, −4.15268174005636343482516316411, −2.67481993328457803333875256532,
0.06092628946667681862994742491, 1.99092005846654008452105236768, 2.91970251105095178939055649214, 4.92799483029777259539538180932, 6.07678337277095492301658788648, 7.33776520335718839501069839874, 7.66621042000989130793759170623, 8.353837230724492393979841007953, 10.15137579155345220576489134144, 11.02671841667348561056326729938