L(s) = 1 | + (0.448 + 0.258i)5-s + (−2.36 + 1.36i)7-s + (−0.189 − 0.328i)11-s + (1.23 − 2.13i)13-s − 3.34i·17-s + 4.73i·19-s + (−0.707 + 1.22i)23-s + (−2.36 − 4.09i)25-s + (−5.91 + 3.41i)29-s + (0.464 + 0.267i)31-s − 1.41·35-s − 4.26·37-s + (4.57 + 2.63i)41-s + (−4.56 + 2.63i)43-s + (4.76 + 8.24i)47-s + ⋯ |
L(s) = 1 | + (0.200 + 0.115i)5-s + (−0.894 + 0.516i)7-s + (−0.0571 − 0.0989i)11-s + (0.341 − 0.591i)13-s − 0.811i·17-s + 1.08i·19-s + (−0.147 + 0.255i)23-s + (−0.473 − 0.819i)25-s + (−1.09 + 0.634i)29-s + (0.0833 + 0.0481i)31-s − 0.239·35-s − 0.701·37-s + (0.713 + 0.412i)41-s + (−0.695 + 0.401i)43-s + (0.694 + 1.20i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.448 - 0.258i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.189 + 0.328i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.23 + 2.13i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.34iT - 17T^{2} \) |
| 19 | \( 1 - 4.73iT - 19T^{2} \) |
| 23 | \( 1 + (0.707 - 1.22i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.91 - 3.41i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.464 - 0.267i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 + (-4.57 - 2.63i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.56 - 2.63i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.76 - 8.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.44iT - 53T^{2} \) |
| 59 | \( 1 + (-6.83 + 11.8i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.59 + 7.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.63 + 5.56i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + (12.6 - 7.29i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.378 + 0.656i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 2.20iT - 89T^{2} \) |
| 97 | \( 1 + (7.19 + 12.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.541191212346546610736504717384, −7.78356254590200289739133293553, −6.97321591310751979798450060764, −5.99104880076370483855852876151, −5.71029158406070230709079070144, −4.55120432158553745777056254034, −3.45293715400910425331042528911, −2.84004280653054999106605338497, −1.62434382095010210384324492643, 0,
1.49917519135787621217683654053, 2.63932492887798242713494506629, 3.73730981570104614282585370647, 4.29715757159106781314908688424, 5.51417993606161408641306706497, 6.13878962693140058265589719440, 7.05498756896310640828254066527, 7.48005574235438953574658177616, 8.817527326129370981337307337787