| L(s) = 1 | + (−3.12 − 9.60i)2-s + (3.16 − 2.30i)3-s + (−56.7 + 41.1i)4-s + (−51.1 − 22.4i)5-s + (−32.0 − 23.2i)6-s + 54.0·7-s + (311. + 226. i)8-s + (−70.3 + 216. i)9-s + (−55.8 + 562. i)10-s + (−193. − 596. i)11-s + (−84.8 + 261. i)12-s + (−53.3 + 164. i)13-s + (−168. − 519. i)14-s + (−213. + 46.7i)15-s + (508. − 1.56e3i)16-s + (−1.06e3 − 775. i)17-s + ⋯ |
| L(s) = 1 | + (−0.551 − 1.69i)2-s + (0.203 − 0.147i)3-s + (−1.77 + 1.28i)4-s + (−0.915 − 0.401i)5-s + (−0.363 − 0.263i)6-s + 0.416·7-s + (1.71 + 1.24i)8-s + (−0.289 + 0.890i)9-s + (−0.176 + 1.77i)10-s + (−0.482 − 1.48i)11-s + (−0.170 + 0.523i)12-s + (−0.0875 + 0.269i)13-s + (−0.230 − 0.708i)14-s + (−0.245 + 0.0536i)15-s + (0.496 − 1.52i)16-s + (−0.896 − 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.237139 + 0.396706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.237139 + 0.396706i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (51.1 + 22.4i)T \) |
| good | 2 | \( 1 + (3.12 + 9.60i)T + (-25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (-3.16 + 2.30i)T + (75.0 - 231. i)T^{2} \) |
| 7 | \( 1 - 54.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + (193. + 596. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (53.3 - 164. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.06e3 + 775. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.78e3 + 1.29e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (230. + 708. i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-5.47e3 + 3.97e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.18e3 + 1.58e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.38e3 + 7.34e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.70e3 - 8.31e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.66e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.60e4 + 1.16e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.82e4 - 2.05e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-8.11e3 + 2.49e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (5.85e3 + 1.80e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.24e4 - 9.06e3i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (4.53e4 - 3.29e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-6.11e3 - 1.88e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (2.65e4 - 1.93e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-5.27e4 - 3.83e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.25e4 + 3.85e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-5.47e4 + 3.97e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−15.97470192352509068058828263039, −13.85851582161262085463701830261, −12.83311819078513013156753320204, −11.35392873780211172613538191445, −10.90338733428297541873938407368, −8.881230520634139361113379391647, −8.084458957930433351418533492167, −4.48220658012285937822425017823, −2.60195525431858699238376318055, −0.36448758856800717254736392658,
4.48077246865025031897157656434, 6.48704262300139103822127157500, 7.70450286904097860767063666747, 8.799367487504510027613869604383, 10.38696700168590971501197541240, 12.45559323482648423292260320442, 14.50706377308672671396690800568, 15.07269102449361913723663888034, 15.85606108239060437100945829455, 17.43330963957620502799470807199
