L(s) = 1 | + (−7.70 − 13.3i)5-s + (−11.9 + 20.6i)7-s + (7.12 − 12.3i)11-s + (6.91 + 11.9i)13-s + 80.5·17-s − 144.·19-s + (−70.5 − 122. i)23-s + (−56.3 + 97.5i)25-s + (−125. + 217. i)29-s + (8.33 + 14.4i)31-s + 367.·35-s + 305.·37-s + (214. + 371. i)41-s + (90.8 − 157. i)43-s + (−39.7 + 68.7i)47-s + ⋯ |
L(s) = 1 | + (−0.689 − 1.19i)5-s + (−0.643 + 1.11i)7-s + (0.195 − 0.338i)11-s + (0.147 + 0.255i)13-s + 1.14·17-s − 1.74·19-s + (−0.639 − 1.10i)23-s + (−0.450 + 0.780i)25-s + (−0.804 + 1.39i)29-s + (0.0482 + 0.0836i)31-s + 1.77·35-s + 1.35·37-s + (0.817 + 1.41i)41-s + (0.322 − 0.557i)43-s + (−0.123 + 0.213i)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
\(L(2)\) |
\(\approx\) |
\(1.272089158\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.272089158\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (7.70 + 13.3i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (11.9 - 20.6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-7.12 + 12.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-6.91 - 11.9i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 80.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (70.5 + 122. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (125. - 217. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-8.33 - 14.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-214. - 371. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-90.8 + 157. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (39.7 - 68.7i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-110. - 190. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-236. + 409. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-323. - 560. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 14.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-128. + 223. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-642. + 1.11e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 156.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (580. - 1.00e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.09643378934956151466863495197, −9.041549114687698390076197525384, −8.639879788404696674270304445809, −7.83033649507918246758272726201, −6.45227540643231598705175701565, −5.69788111212983195763211504583, −4.62775463683312949195832914169, −3.68680962001030151557493391691, −2.33543601761627608370671790317, −0.793626959977809006047739583763,
0.53128270204561195237311779156, 2.36759177346726784238267057638, 3.72299854824730308039938188649, 4.01817254202835137098100178125, 5.80469989618407623073892996550, 6.67627491874674725461919660044, 7.46791649050064600639478226561, 8.018909622352849352733845016592, 9.514377442692987879787932443586, 10.21411681557275760664496909118