L(s) = 1 | − 5.29i·3-s − 8·7-s − 1.00·9-s − 15.8i·11-s + 52.9i·13-s + 14·17-s − 37.0i·19-s + 42.3i·21-s − 152·23-s − 137. i·27-s + 158. i·29-s − 224·31-s − 84.0·33-s + 243. i·37-s + 280·39-s + ⋯ |
L(s) = 1 | − 1.01i·3-s − 0.431·7-s − 0.0370·9-s − 0.435i·11-s + 1.12i·13-s + 0.199·17-s − 0.447i·19-s + 0.439i·21-s − 1.37·23-s − 0.980i·27-s + 1.01i·29-s − 1.29·31-s − 0.443·33-s + 1.08i·37-s + 1.14·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9017986903\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9017986903\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 5.29iT - 27T^{2} \) |
| 7 | \( 1 + 8T + 343T^{2} \) |
| 11 | \( 1 + 15.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 52.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 14T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 152T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 336T + 1.03e5T^{2} \) |
| 53 | \( 1 - 31.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 534. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 95.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 174. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 72T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294T + 3.89e5T^{2} \) |
| 79 | \( 1 - 464T + 4.93e5T^{2} \) |
| 83 | \( 1 - 545. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 266T + 7.04e5T^{2} \) |
| 97 | \( 1 + 994T + 9.12e5T^{2} \) |
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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
−9.945180541227773339756653636036, −9.157821486564181833801476207821, −8.223625288263474519782614322533, −7.34953398701924564583947121570, −6.63231441976888985963369956135, −5.93434359289415222397581019190, −4.62120747926129853004260095382, −3.49601985253720186966975684082, −2.20252078332611003468659867997, −1.20275084940894231248047869013,
0.24853484667633110509320176956, 2.05035848570045410945505335531, 3.46771193590558800220132001357, 4.08533632653934119034433227853, 5.24718275168003991439486415249, 5.96289589358899119272046804422, 7.22270888911280730521447734003, 8.026227797089654370630206593256, 9.093052370770075259444537916697, 9.898476345477803808246889285268