L(s) = 1 | + (−2.82 − 2.82i)2-s + 16.0i·4-s + (−158. + 158. i)7-s + (45.2 − 45.2i)8-s − 147. i·11-s + (−516. − 516. i)13-s + 897.·14-s − 256.·16-s + (1.09e3 + 1.09e3i)17-s + 1.17e3i·19-s + (−415. + 415. i)22-s + (−3.13e3 + 3.13e3i)23-s + 2.92e3i·26-s + (−2.53e3 − 2.53e3i)28-s − 1.86e3·29-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−1.22 + 1.22i)7-s + (0.250 − 0.250i)8-s − 0.366i·11-s + (−0.847 − 0.847i)13-s + 1.22·14-s − 0.250·16-s + (0.917 + 0.917i)17-s + 0.747i·19-s + (−0.183 + 0.183i)22-s + (−1.23 + 1.23i)23-s + 0.847i·26-s + (−0.611 − 0.611i)28-s − 0.411·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.3701641456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3701641456\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (158. - 158. i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 147. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (516. + 516. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.09e3 - 1.09e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 1.17e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (3.13e3 - 3.13e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 1.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (1.68e3 - 1.68e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 9.09e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.37e4 - 1.37e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (887. + 887. i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.63e4 - 1.63e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 957.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-3.57e4 + 3.57e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 2.59e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (5.28e4 + 5.28e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 1.63e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (6.95e4 - 6.95e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.33e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-8.35e3 + 8.35e3i)T - 8.58e9iT^{2} \) |
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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
−9.839533635552824611284522346049, −9.404726777860293068204638351176, −8.281745504135455159879342786452, −7.53449527627811987077716176225, −6.07208583231856365537526302651, −5.54052854964924346285975866517, −3.70888855593543488157618016481, −2.94890487393481971471669676798, −1.77855046473393621552508729391, −0.14781556920054741482399153619,
0.73422995956783340301978322409, 2.38913854716252843117866922462, 3.80357575272277004455390073675, 4.82939787062503387689958212925, 6.17343217397259023833539261841, 7.08871483074540468125732540848, 7.46600177622078619337515627909, 8.858588157116502710618363526149, 9.850953414824646893703168319190, 10.06500878543586122393648490110