L(s) = 1 | + (−5.5 − 9.52i)2-s + (−44.5 + 77.0i)4-s + 626.·8-s + (121.5 + 210. i)9-s + (38 − 65.8i)11-s + (−2.02e3 − 3.50e3i)16-s + (1.33e3 − 2.31e3i)18-s − 836·22-s + (2.47e3 + 4.28e3i)23-s + (1.56e3 − 2.70e3i)25-s + 7.28e3·29-s + (−1.22e4 + 2.11e4i)32-s − 2.16e4·36-s + (4.44e3 + 7.69e3i)37-s + 1.17e4·43-s + (3.38e3 + 5.85e3i)44-s + ⋯ |
L(s) = 1 | + (−0.972 − 1.68i)2-s + (−1.39 + 2.40i)4-s + 3.46·8-s + (0.5 + 0.866i)9-s + (0.0946 − 0.164i)11-s + (−1.97 − 3.42i)16-s + (0.972 − 1.68i)18-s − 0.368·22-s + (0.975 + 1.69i)23-s + (0.5 − 0.866i)25-s + 1.60·29-s + (−2.11 + 3.65i)32-s − 2.78·36-s + (0.533 + 0.924i)37-s + 0.968·43-s + (0.263 + 0.456i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.810875 - 0.401972i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.810875 - 0.401972i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (5.5 + 9.52i)T + (-16 + 27.7i)T^{2} \) |
| 3 | \( 1 + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-38 + 65.8i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 3.71e5T^{2} \) |
| 17 | \( 1 + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.47e3 - 4.28e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 7.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-4.44e3 - 7.69e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.22e4 - 2.12e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.46e4 - 6.00e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.22e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.00e4 + 6.94e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.83402098173615237309563946479, −12.94075601727643739658634060184, −11.80890450617666156216612863444, −10.79188271225877721964154660048, −9.889417980539586984598260024936, −8.676409011842994005936267099199, −7.50592385933956759888849705418, −4.54420280589619847263599005074, −2.85586637449605103085036914909, −1.23887419323159435713920577928,
0.817019250532297601689552575162, 4.68982984388865013993011455444, 6.28676626806682348573626347119, 7.17909803497092025660594220749, 8.571915459378809512632589738251, 9.499418099516848113232982554769, 10.66600418153605608773764310597, 12.76032305916006270488256223925, 14.25487887236242164915534054361, 15.02803088170051434083531658119