L(s) = 1 | + (3.20e4 − 5.71e4i)2-s + 1.43e7i·3-s + (−2.24e9 − 3.66e9i)4-s − 4.45e10·5-s + (8.19e11 + 4.58e11i)6-s − 4.42e13i·7-s + (−2.81e14 + 1.12e13i)8-s + 1.64e15·9-s + (−1.42e15 + 2.54e15i)10-s + 2.28e16i·11-s + (5.24e16 − 3.21e16i)12-s − 7.09e17·13-s + (−2.53e18 − 1.41e18i)14-s − 6.38e17i·15-s + (−8.36e18 + 1.64e19i)16-s − 5.13e19·17-s + ⋯ |
L(s) = 1 | + (0.488 − 0.872i)2-s + 0.332i·3-s + (−0.522 − 0.852i)4-s − 0.292·5-s + (0.290 + 0.162i)6-s − 1.33i·7-s + (−0.999 + 0.0398i)8-s + 0.889·9-s + (−0.142 + 0.254i)10-s + 0.497i·11-s + (0.283 − 0.174i)12-s − 1.06·13-s + (−1.16 − 0.650i)14-s − 0.0972i·15-s + (−0.453 + 0.891i)16-s − 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(0.3684140492\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3684140492\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.20e4 + 5.71e4i)T \) |
good | 3 | \( 1 - 1.43e7iT - 1.85e15T^{2} \) |
| 5 | \( 1 + 4.45e10T + 2.32e22T^{2} \) |
| 7 | \( 1 + 4.42e13iT - 1.10e27T^{2} \) |
| 11 | \( 1 - 2.28e16iT - 2.11e33T^{2} \) |
| 13 | \( 1 + 7.09e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + 5.13e19T + 2.36e39T^{2} \) |
| 19 | \( 1 - 9.08e19iT - 8.31e40T^{2} \) |
| 23 | \( 1 - 3.40e21iT - 3.76e43T^{2} \) |
| 29 | \( 1 + 9.75e22T + 6.26e46T^{2} \) |
| 31 | \( 1 - 8.90e23iT - 5.29e47T^{2} \) |
| 37 | \( 1 + 1.36e25T + 1.52e50T^{2} \) |
| 41 | \( 1 - 9.45e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + 2.05e26iT - 1.86e52T^{2} \) |
| 47 | \( 1 + 6.70e25iT - 3.21e53T^{2} \) |
| 53 | \( 1 + 3.12e27T + 1.50e55T^{2} \) |
| 59 | \( 1 + 4.03e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 4.61e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + 2.83e29iT - 2.71e58T^{2} \) |
| 71 | \( 1 - 3.78e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + 8.28e29T + 4.22e59T^{2} \) |
| 79 | \( 1 - 3.22e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 - 3.96e29iT - 2.57e61T^{2} \) |
| 89 | \( 1 - 1.44e31T + 2.40e62T^{2} \) |
| 97 | \( 1 - 5.69e30T + 3.77e63T^{2} \) |
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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
−15.51448269627413244007352333110, −13.92757792407982489094629824349, −12.50423595156291650410879589728, −10.76901329606727624879236541991, −9.679363736347633408081995486239, −7.16408161032747303087758809670, −4.73726168667176252870942628342, −3.75608466357675924408105734550, −1.73766525367221020102943367279, −0.098759474866183931409339251022,
2.48502706778180948986390503951, 4.48174841583370276438922541197, 6.07113246530241338367473689295, 7.59448601685614221114195287210, 9.140334657735479148115134412393, 11.92401247100302308911426330110, 13.12082649214843069216337643384, 14.93881869860480484744171579604, 15.96706766780928901889923361941, 17.77574858149185229453587528543