L(s) = 1 | − 10.2i·2-s − 5.52i·3-s − 73.3·4-s − 56.6·6-s − 68.9i·7-s + 423. i·8-s + 212.·9-s − 486.·11-s + 404. i·12-s − 428. i·13-s − 707.·14-s + 2.00e3·16-s − 1.80e3i·17-s − 2.18e3i·18-s + 1.04e3·19-s + ⋯ |
L(s) = 1 | − 1.81i·2-s − 0.354i·3-s − 2.29·4-s − 0.642·6-s − 0.531i·7-s + 2.34i·8-s + 0.874·9-s − 1.21·11-s + 0.811i·12-s − 0.703i·13-s − 0.964·14-s + 1.95·16-s − 1.51i·17-s − 1.58i·18-s + 0.665·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.258540 + 1.09519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.258540 + 1.09519i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + 10.2iT - 32T^{2} \) |
| 3 | \( 1 + 5.52iT - 243T^{2} \) |
| 7 | \( 1 + 68.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 486.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 428. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.80e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 686. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.97e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.50e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 895. iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.71e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.37e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.29e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5iT - 8.58e9T^{2} \) |
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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
−15.83915321866297565071725416832, −13.76460678795824617108157744179, −13.07250705823983688589885200640, −11.90523044566214385587897925862, −10.57556128662513838306914408334, −9.672876579993069266008313221120, −7.72132452802252651792505504663, −4.76564772446388301793929903484, −2.84117052438160518418823621882, −0.830399145543979191803325939975,
4.50796753060133859929917454680, 5.96381000751118474725766750654, 7.48600928504577769048600575301, 8.764450992946443129523729325976, 10.20211745679126918043741198789, 12.66477025061461029138078508074, 13.89973710065286387148813923578, 15.29199420958373474019884720908, 15.75470738141161281502677871300, 16.90571420575665478102109121817