L(s) = 1 | − 0.506i·3-s + 5.30i·5-s + 8.74·9-s + 17.0·11-s + 21.4i·13-s + 2.68·15-s − 24.0i·17-s − 12.2i·19-s + 40.2·23-s − 3.11·25-s − 8.99i·27-s − 26.0·29-s + 25.2i·31-s − 8.66i·33-s + 12.9·37-s + ⋯ |
L(s) = 1 | − 0.168i·3-s + 1.06i·5-s + 0.971·9-s + 1.55·11-s + 1.65i·13-s + 0.179·15-s − 1.41i·17-s − 0.643i·19-s + 1.75·23-s − 0.124·25-s − 0.333i·27-s − 0.899·29-s + 0.814i·31-s − 0.262i·33-s + 0.350·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.573798072\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.573798072\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.506iT - 9T^{2} \) |
| 5 | \( 1 - 5.30iT - 25T^{2} \) |
| 11 | \( 1 - 17.0T + 121T^{2} \) |
| 13 | \( 1 - 21.4iT - 169T^{2} \) |
| 17 | \( 1 + 24.0iT - 289T^{2} \) |
| 19 | \( 1 + 12.2iT - 361T^{2} \) |
| 23 | \( 1 - 40.2T + 529T^{2} \) |
| 29 | \( 1 + 26.0T + 841T^{2} \) |
| 31 | \( 1 - 25.2iT - 961T^{2} \) |
| 37 | \( 1 - 12.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.72T + 2.80e3T^{2} \) |
| 59 | \( 1 - 49.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.93iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 105.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 35.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 46.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 86.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 64.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 42.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 28.7iT - 9.40e3T^{2} \) |
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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
−9.270811268229954371839897873320, −8.845522504617096572257486833627, −7.18119569780022166520356991540, −6.98432187775268850569612657693, −6.57618989429645006910541016796, −5.09603413235126942075674339150, −4.21991188928832077422624108505, −3.34865683706350882320007957308, −2.19525039179471744862896216783, −1.10018686681295762715658794370,
0.914953137891084150327055173891, 1.58996917410598805725031735199, 3.33662156794051853534453554649, 4.10894564803466221634957787451, 4.93552649496519718863669784706, 5.83441197208157192560669757092, 6.67135375062244985550593382348, 7.72820172918423222862099785423, 8.381313384175493699047433338393, 9.219303745669668562947603962045