| L(s) = 1 | + (14.9 − 22.4i)3-s + (202. + 116. i)5-s + (−95.5 − 165. i)7-s + (−282. − 672. i)9-s + (673. − 388. i)11-s + (45.5 − 78.9i)13-s + (5.64e3 − 2.80e3i)15-s − 7.04e3i·17-s − 2.73e3·19-s + (−5.15e3 − 323. i)21-s + (1.72e4 + 9.94e3i)23-s + (1.94e4 + 3.37e4i)25-s + (−1.93e4 − 3.68e3i)27-s + (2.71e4 − 1.56e4i)29-s + (−6.17e3 + 1.06e4i)31-s + ⋯ |
| L(s) = 1 | + (0.553 − 0.832i)3-s + (1.61 + 0.934i)5-s + (−0.278 − 0.482i)7-s + (−0.387 − 0.921i)9-s + (0.505 − 0.291i)11-s + (0.0207 − 0.0359i)13-s + (1.67 − 0.830i)15-s − 1.43i·17-s − 0.398·19-s + (−0.556 − 0.0349i)21-s + (1.41 + 0.817i)23-s + (1.24 + 2.15i)25-s + (−0.982 − 0.187i)27-s + (1.11 − 0.641i)29-s + (−0.207 + 0.358i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.268969899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.268969899\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-14.9 + 22.4i)T \) |
| good | 5 | \( 1 + (-202. - 116. i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (95.5 + 165. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-673. + 388. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-45.5 + 78.9i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 7.04e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.73e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.72e4 - 9.94e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.71e4 + 1.56e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (6.17e3 - 1.06e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.79e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (3.74e4 + 2.16e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (1.92e4 + 3.33e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.43e5 + 8.30e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 5.47e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.41e4 - 8.14e3i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.94e4 - 5.09e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.47e5 + 2.56e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 1.57e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 8.02e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + (1.88e5 + 3.26e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (7.33e5 - 4.23e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + 1.12e3iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-6.75e5 - 1.16e6i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−11.87824421096585238912239235300, −10.67025683126140483029696889692, −9.649659729172599161880099022886, −8.859068801479102609848154609432, −7.13128395894982018335040346172, −6.69330793613183205760110174200, −5.48337813838052112142657088613, −3.28661617214668128578129176998, −2.32279198554049567708275168680, −1.00252922630380781280717327263,
1.47842335846512345503048432330, 2.67283269665041489150581884568, 4.38635622048700070786403108158, 5.41958546630705798500141515065, 6.45583204490379204344336004519, 8.541665534908672492200967703785, 8.998682138971066771857416152037, 9.932232322237776506500511890330, 10.71909269561098254626210926081, 12.47411067241190766411521217532
