| L(s) = 1 | + (1.53 + 1.28i)2-s + (−2.81 + 1.02i)3-s + (0.694 + 3.93i)4-s + (−2.27 + 12.9i)5-s + (−5.63 − 2.05i)6-s + (−8.34 − 14.4i)7-s + (−4.00 + 6.92i)8-s + (6.89 − 5.78i)9-s + (−20.0 + 16.8i)10-s + (−18.8 + 32.6i)11-s + (−6 − 10.3i)12-s + (−60.4 − 21.9i)13-s + (5.79 − 32.8i)14-s + (−6.83 − 38.7i)15-s + (−15.0 + 5.47i)16-s + (−43.8 − 36.8i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.542 + 0.197i)3-s + (0.0868 + 0.492i)4-s + (−0.203 + 1.15i)5-s + (−0.383 − 0.139i)6-s + (−0.450 − 0.780i)7-s + (−0.176 + 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.635 + 0.533i)10-s + (−0.516 + 0.894i)11-s + (−0.144 − 0.249i)12-s + (−1.28 − 0.469i)13-s + (0.110 − 0.627i)14-s + (−0.117 − 0.667i)15-s + (−0.234 + 0.0855i)16-s + (−0.626 − 0.525i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0448270 + 0.949570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0448270 + 0.949570i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.53 - 1.28i)T \) |
| 3 | \( 1 + (2.81 - 1.02i)T \) |
| 19 | \( 1 + (-33.3 - 75.8i)T \) |
| good | 5 | \( 1 + (2.27 - 12.9i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (8.34 + 14.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (18.8 - 32.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (60.4 + 21.9i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (43.8 + 36.8i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-30.3 - 172. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-45.0 + 37.7i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-4.64 - 8.03i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 21.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-380. + 138. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (46.1 - 261. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (11.3 - 9.50i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-94.4 - 535. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-339. - 284. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-81.3 - 461. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (597. - 501. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-100. + 571. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-240. + 87.3i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (1.25e3 - 455. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (496. + 859. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-738. - 268. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (1.05e3 + 883. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−13.69045339170822235021207737887, −12.62649143772479793522935874363, −11.59695582172338653390328801195, −10.47330292307332556340647245829, −9.710036710927221099826376750579, −7.39505837553707221504850129852, −7.19821457740567504339760210952, −5.69301445410359157465924548011, −4.34567462190152722661128664155, −2.90619004106848331819660701051,
0.43903945143011044817395276315, 2.52943835461818877484820711040, 4.55178500666692961625294756456, 5.41269450327607324820002047164, 6.69389203078426556380678385377, 8.438742520111946363854482421995, 9.406915101368133287143132639251, 10.77539576629490496908991826387, 11.84786514982133032182472623717, 12.60320963661065589531713132143
