L(s) = 1 | + (2.62 − 4.54i)2-s + (−9.74 − 16.8i)4-s + (−2.03 − 3.51i)5-s + (3.5 − 6.06i)7-s − 60.2·8-s − 21.2·10-s + (−7.57 + 13.1i)11-s + (−24.2 − 42.0i)13-s + (−18.3 − 31.7i)14-s + (−80.0 + 138. i)16-s − 107.·17-s + 109.·19-s + (−39.5 + 68.5i)20-s + (39.7 + 68.8i)22-s + (92.3 + 159. i)23-s + ⋯ |
L(s) = 1 | + (0.926 − 1.60i)2-s + (−1.21 − 2.11i)4-s + (−0.181 − 0.314i)5-s + (0.188 − 0.327i)7-s − 2.66·8-s − 0.673·10-s + (−0.207 + 0.359i)11-s + (−0.518 − 0.897i)13-s + (−0.350 − 0.606i)14-s + (−1.25 + 2.16i)16-s − 1.53·17-s + 1.32·19-s + (−0.442 + 0.766i)20-s + (0.384 + 0.666i)22-s + (0.837 + 1.44i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.690527 + 1.81011i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.690527 + 1.81011i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-3.5 + 6.06i)T \) |
good | 2 | \( 1 + (-2.62 + 4.54i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (2.03 + 3.51i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (7.57 - 13.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (24.2 + 42.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-92.3 - 159. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-47.0 + 81.4i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (67.6 + 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (148. + 258. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-193. + 334. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (36.7 - 63.6i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 633.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (162. + 281. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-34.7 + 60.2i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-139. - 242. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 497.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 457.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-548. + 949. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-39.3 + 68.1i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 292.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (82.2 - 142. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.62810517231761607484127312636, −10.75908176256440248380104531341, −9.915395684358842891374980485889, −8.914527529565292412709594883862, −7.31454989437977748150013560717, −5.51263066634268295138994185775, −4.69943363266976151062684382198, −3.51916247805068360018953952556, −2.21605179294778495611864595173, −0.63504101677335192491793836038,
2.98111015325823088279034742187, 4.45890185756917765792410508055, 5.28784945896513681885268158694, 6.63557991126300018954369516796, 7.15752287728600056225936087117, 8.439041856234090756459459518025, 9.188520839142972476863580298416, 11.00652650660538849658018273919, 12.07917230417496472483589678157, 13.04843426905374747523210733018