L(s) = 1 | + (1.12 − 0.649i)2-s + (−1.28 − 5.03i)3-s + (−3.15 + 5.46i)4-s + (−4.00 − 2.31i)5-s + (−4.71 − 4.83i)6-s + (−7.19 + 4.15i)7-s + 18.5i·8-s + (−23.7 + 12.9i)9-s − 6.01·10-s + (−32.0 + 18.5i)11-s + (31.5 + 8.87i)12-s + (46.3 + 7.14i)13-s + (−5.39 + 9.35i)14-s + (−6.51 + 23.1i)15-s + (−13.1 − 22.7i)16-s − 98.9·17-s + ⋯ |
L(s) = 1 | + (0.397 − 0.229i)2-s + (−0.246 − 0.969i)3-s + (−0.394 + 0.683i)4-s + (−0.358 − 0.207i)5-s + (−0.320 − 0.328i)6-s + (−0.388 + 0.224i)7-s + 0.821i·8-s + (−0.878 + 0.478i)9-s − 0.190·10-s + (−0.879 + 0.507i)11-s + (0.759 + 0.213i)12-s + (0.988 + 0.152i)13-s + (−0.103 + 0.178i)14-s + (−0.112 + 0.398i)15-s + (−0.205 − 0.356i)16-s − 1.41·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.197042 + 0.340023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.197042 + 0.340023i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.28 + 5.03i)T \) |
| 13 | \( 1 + (-46.3 - 7.14i)T \) |
good | 2 | \( 1 + (-1.12 + 0.649i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (4.00 + 2.31i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (7.19 - 4.15i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (32.0 - 18.5i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + 98.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 138. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-35.5 + 61.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 3.03i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (109. + 63.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 256. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (390. + 225. i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (202. + 351. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-498. + 287. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (177. + 102. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-171. - 297. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-451. - 260. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 294. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 539. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-234. - 406. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (299. - 173. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.09e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (482. - 278. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.24620531956062790829046232892, −12.43263285023585015186915610352, −11.76949065298566969006478055228, −10.54234726011777194947039951495, −8.710936539378862516339996088152, −8.070129657840092598226394671036, −6.78174411394034048647194926112, −5.40336600569771018497825266609, −3.89235809561801459602092817294, −2.26624594000930280783716486891,
0.18288904008966978606921616003, 3.34627106707242235330967156859, 4.59105445707246226877562026039, 5.65117982797898096296733442061, 6.81855092977741220413765249546, 8.680251114775204529635249296692, 9.545399814540101162382955640469, 10.83854655930070543945023170530, 11.17646587856897500352865150000, 13.15296266498510201423324589537