| L(s) = 1 | + (−1.74 − 5.36i)2-s + (−7.28 − 5.29i)3-s + (0.152 − 0.110i)4-s + (21.0 − 64.7i)5-s + (−15.6 + 48.2i)6-s + (−106. + 77.4i)7-s + (−146. − 106. i)8-s + (25.0 + 77.0i)9-s − 384.·10-s + (−98.6 + 388. i)11-s − 1.69·12-s + (−4.06 − 12.5i)13-s + (600. + 436. i)14-s + (−496. + 360. i)15-s + (−314. + 968. i)16-s + (618. − 1.90e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.308 − 0.948i)2-s + (−0.467 − 0.339i)3-s + (0.00476 − 0.00346i)4-s + (0.376 − 1.15i)5-s + (−0.177 + 0.547i)6-s + (−0.821 + 0.597i)7-s + (−0.811 − 0.589i)8-s + (0.103 + 0.317i)9-s − 1.21·10-s + (−0.245 + 0.969i)11-s − 0.00339·12-s + (−0.00667 − 0.0205i)13-s + (0.819 + 0.595i)14-s + (−0.569 + 0.413i)15-s + (−0.307 + 0.945i)16-s + (0.519 − 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.290i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.123662 + 0.834458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123662 + 0.834458i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.28 + 5.29i)T \) |
| 11 | \( 1 + (98.6 - 388. i)T \) |
| good | 2 | \( 1 + (1.74 + 5.36i)T + (-25.8 + 18.8i)T^{2} \) |
| 5 | \( 1 + (-21.0 + 64.7i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (106. - 77.4i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (4.06 + 12.5i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-618. + 1.90e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (2.25e3 + 1.63e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 2.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-1.92e3 + 1.40e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (603. + 1.85e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-7.39e3 + 5.37e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (7.64e3 + 5.55e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (9.32e3 + 6.77e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (5.78e3 + 1.77e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.67e4 + 2.67e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.06e4 - 3.26e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 3.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (7.49e3 - 2.30e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.77e4 + 3.47e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.30e4 - 4.00e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-5.75e3 + 1.77e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 8.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.62e4 - 1.11e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
| 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
−15.26508898970995673787067283209, −13.05423431652118037200037720995, −12.55714894480814100872919919350, −11.41518000630249905578047495748, −9.870056178306505187238023416324, −9.017481002719024985166766864063, −6.73788899602552243658016555929, −5.09837921219479212205065532068, −2.39152381824872033114596091198, −0.57332494550074734156105506954,
3.29992255338462840500113835969, 6.04387955383525295037384864697, 6.68802934701342808366375841917, 8.319622884941196527539879323349, 10.16575229954841573722497579385, 11.01402761161117412472496077872, 12.79421818857777573840824813942, 14.41373836204648274529925302504, 15.24602334497214476929415451021, 16.61243710162430495993512487724
