L(s) = 1 | + (−9.08 − 15.7i)5-s + (0.411 − 0.712i)7-s + (32.7 − 56.7i)11-s + (−15.9 − 27.5i)13-s + 124.·17-s + 27.8·19-s + (21.5 + 37.3i)23-s + (−102. + 177. i)25-s + (−19.9 + 34.4i)29-s + (−147. − 255. i)31-s − 14.9·35-s − 104.·37-s + (−153. − 266. i)41-s + (180. − 312. i)43-s + (−198. + 344. i)47-s + ⋯ |
L(s) = 1 | + (−0.812 − 1.40i)5-s + (0.0222 − 0.0384i)7-s + (0.898 − 1.55i)11-s + (−0.339 − 0.587i)13-s + 1.77·17-s + 0.336·19-s + (0.195 + 0.338i)23-s + (−0.821 + 1.42i)25-s + (−0.127 + 0.220i)29-s + (−0.854 − 1.47i)31-s − 0.0722·35-s − 0.462·37-s + (−0.585 − 1.01i)41-s + (0.640 − 1.10i)43-s + (−0.617 + 1.06i)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
\(L(2)\) |
\(\approx\) |
\(1.379923460\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.379923460\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (9.08 + 15.7i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-0.411 + 0.712i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-32.7 + 56.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (15.9 + 27.5i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-21.5 - 37.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (19.9 - 34.4i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (147. + 255. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 104.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (153. + 266. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-180. + 312. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (198. - 344. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (94.4 + 163. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-49.5 + 85.7i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (212. + 368. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 445.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 499.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (285. - 493. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (654. - 1.13e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (472. - 818. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−9.525835108507050014553333527649, −8.873005279105180215585068595377, −8.056512181064439179085184743951, −7.42422173343910784535650545228, −5.82883420692286476517917298729, −5.30925894305062834114222246171, −4.00865596334124051192012362529, −3.29852017618117473289401513329, −1.24831796389745582490219618672, −0.44765400869901656363301805326,
1.60962688880860810806467503402, 2.99979474802813970241092116733, 3.84905901404761364164469600272, 4.92506044821481245064456174046, 6.33031513981930267417000824037, 7.23770839781293046169612922084, 7.47757213934273766017662799180, 8.837520053457757097287001682790, 9.966929616774174901421752228909, 10.33413478571631785135283595580