L(s) = 1 | − 10.7i·3-s + 25i·5-s − 198.·7-s + 127.·9-s − 85.9i·11-s + 407. i·13-s + 268.·15-s + 1.20e3·17-s + 206. i·19-s + 2.13e3i·21-s + 2.59e3·23-s − 625·25-s − 3.98e3i·27-s + 6.19e3i·29-s + 1.86e3·31-s + ⋯ |
L(s) = 1 | − 0.689i·3-s + 0.447i·5-s − 1.53·7-s + 0.524·9-s − 0.214i·11-s + 0.668i·13-s + 0.308·15-s + 1.01·17-s + 0.130i·19-s + 1.05i·21-s + 1.02·23-s − 0.200·25-s − 1.05i·27-s + 1.36i·29-s + 0.348·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.667163062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.667163062\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
good | 3 | \( 1 + 10.7iT - 243T^{2} \) |
| 7 | \( 1 + 198.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 85.9iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 407. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 206. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.19e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.47e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.26e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.07e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.13e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.26e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.80e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.13e5T + 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.33859847989328198086651597340, −10.94802593519368552729879820308, −9.912202452287071334504511577129, −9.040371988768907034432907372800, −7.43421330043882178702813393979, −6.81202817738313948033692251719, −5.78174481872371555955247518178, −3.88348320006659982165394315419, −2.66962030976711119621374947476, −0.949451348844032444099508165619,
0.76310990282658015833754615043, 2.95057851242125985210581669047, 4.07282944184490804779821144625, 5.37095869104114946240195014938, 6.58441175179788059363841361262, 7.83213943308547388004150314474, 9.311096869359012516996615088728, 9.811180884866177018681842435054, 10.70295865660830849785010673604, 12.23147225659268308547253070728