L(s) = 1 | − 7.81i·2-s + 5.19·3-s − 45.1·4-s + 21.3·5-s − 40.6i·6-s − 30.7·7-s + 227. i·8-s + 27·9-s − 166. i·10-s + 207. i·11-s − 234.·12-s + 206. i·13-s + 240. i·14-s + 110.·15-s + 1.05e3·16-s − 391.·17-s + ⋯ |
L(s) = 1 | − 1.95i·2-s + 0.577·3-s − 2.82·4-s + 0.852·5-s − 1.12i·6-s − 0.627·7-s + 3.55i·8-s + 0.333·9-s − 1.66i·10-s + 1.71i·11-s − 1.62·12-s + 1.21i·13-s + 1.22i·14-s + 0.492·15-s + 4.13·16-s − 1.35·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 + 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.109860004\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109860004\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (3.46e3 + 327. i)T \) |
good | 2 | \( 1 + 7.81iT - 16T^{2} \) |
| 5 | \( 1 - 21.3T + 625T^{2} \) |
| 7 | \( 1 + 30.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 207. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 206. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 391.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 321.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 287. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.02e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 560. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.25e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.52e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.15e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 245. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.14e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 518. iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.49e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 9.31e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.00e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 5.13e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.34e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.14e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.32e3iT - 8.85e7T^{2} \) |
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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
−12.14095104589940610231984535134, −10.90515124632062156372725912313, −9.900646746388006524715874260998, −9.467305258614447593409102799116, −8.612622741198397256804141332025, −6.74871267629684733939794495116, −4.82559824854618155285620764722, −3.93029890145078721301510822413, −2.33476818771027573391606817298, −1.81786551632156286109827105068,
0.37375963864992062320504644804, 3.22905699916780354865536122876, 4.76153810735689083761831655047, 6.14425770340172358910647443572, 6.41677497575799181450289999712, 8.036188241220763555864720810008, 8.604019689744277299474921746278, 9.525638657648156727439768284341, 10.54136048661674832408338399388, 12.84385634505631471724409649663