L(s) = 1 | + (−2.5 − 4.33i)5-s + (−0.606 + 1.05i)7-s + (−16.8 + 29.1i)11-s + (−38.7 − 67.1i)13-s + 73.2·17-s + 39.2·19-s + (−17.6 − 30.4i)23-s + (−12.5 + 21.6i)25-s + (7.89 − 13.6i)29-s + (−26.1 − 45.3i)31-s + 6.06·35-s − 67.1·37-s + (12.5 + 21.6i)41-s + (15.8 − 27.4i)43-s + (206. − 357. i)47-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (−0.0327 + 0.0567i)7-s + (−0.460 + 0.797i)11-s + (−0.827 − 1.43i)13-s + 1.04·17-s + 0.473·19-s + (−0.159 − 0.276i)23-s + (−0.100 + 0.173i)25-s + (0.0505 − 0.0875i)29-s + (−0.151 − 0.262i)31-s + 0.0292·35-s − 0.298·37-s + (0.0476 + 0.0826i)41-s + (0.0562 − 0.0974i)43-s + (0.639 − 1.10i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2330942968\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2330942968\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 7 | \( 1 + (0.606 - 1.05i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (16.8 - 29.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (38.7 + 67.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 73.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (17.6 + 30.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-7.89 + 13.6i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (26.1 + 45.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 67.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-12.5 - 21.6i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-15.8 + 27.4i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-206. + 357. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 54.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + (257. + 446. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (431. - 747. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-183. - 318. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 797.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-42.7 + 74.1i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (45.9 - 79.6i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 888.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (166. - 288. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−9.418054931140620252262129305986, −8.453024836636088153207921839318, −7.62113289051282871346317513766, −7.28957603018696472065426825314, −5.87199535325079719665074946976, −5.27633424001799195319240605867, −4.45529678091106265882539483537, −3.30550753788332429831220007853, −2.42541926091310421797203343965, −1.07127249158505783498009172049,
0.05518089452995391561746595447, 1.46251935825834307852317716759, 2.68198264763332704239522575728, 3.52642814685450245174741473598, 4.52888604856939398869501048344, 5.46742922412066870841417588393, 6.30783317722321553543751216477, 7.26286368192055757296234525219, 7.75113162365996854142246001818, 8.788437964015849897737645644445