L(s) = 1 | + (−6.67 − 11.5i)5-s + (7.17 − 12.4i)7-s + (−19.5 + 33.8i)11-s + (38.3 + 66.5i)13-s − 62.4·17-s − 39.7·19-s + (64.2 + 111. i)23-s + (−26.7 + 46.2i)25-s + (−32.4 + 56.2i)29-s + (4.56 + 7.91i)31-s − 191.·35-s + 319.·37-s + (−8.78 − 15.2i)41-s + (225. − 390. i)43-s + (290. − 503. i)47-s + ⋯ |
L(s) = 1 | + (−0.597 − 1.03i)5-s + (0.387 − 0.671i)7-s + (−0.535 + 0.927i)11-s + (0.819 + 1.41i)13-s − 0.890·17-s − 0.480·19-s + (0.582 + 1.00i)23-s + (−0.213 + 0.370i)25-s + (−0.207 + 0.360i)29-s + (0.0264 + 0.0458i)31-s − 0.926·35-s + 1.41·37-s + (−0.0334 − 0.0579i)41-s + (0.799 − 1.38i)43-s + (0.902 − 1.56i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.575485235\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.575485235\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (6.67 + 11.5i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-7.17 + 12.4i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (19.5 - 33.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-38.3 - 66.5i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 62.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-64.2 - 111. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (32.4 - 56.2i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-4.56 - 7.91i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (8.78 + 15.2i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-225. + 390. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-290. + 503. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-120. - 209. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (248. - 430. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-289. - 500. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 660.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (365. - 632. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (545. - 944. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-742. + 1.28e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.24740728663355213102222850217, −9.075396148973062908129299322099, −8.641493918465976066654572203276, −7.50852177902513863342882901973, −6.88832444732869969507061377588, −5.49233250671400744812954898541, −4.37007991785573504338928551822, −4.06976479400176359727968306524, −2.12049258378068866453526584073, −0.946118877479188781225987376750,
0.58082202613745626440706377920, 2.53245855874569179348782902801, 3.21251882227317435614488628154, 4.48377993178721356262317994049, 5.76850723464646742918208016649, 6.39529178170485757368413336940, 7.66025664736748475624700253478, 8.245225544143994473022391668124, 9.074787372990720362230788355743, 10.47176825520068441720946429014