| L(s) = 1 | + (1.53 + 1.28i)2-s + (0.694 + 3.93i)4-s + (7.46 + 2.71i)5-s + (−2.66 + 15.0i)7-s + (−4.00 + 6.92i)8-s + (7.94 + 13.7i)10-s + (22.9 − 8.33i)11-s + (−33.4 + 28.0i)13-s + (−23.4 + 19.6i)14-s + (−15.0 + 5.47i)16-s + (46.3 + 80.3i)17-s + (13.1 − 22.7i)19-s + (−5.51 + 31.2i)20-s + (45.8 + 16.6i)22-s + (11.6 + 65.9i)23-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.0868 + 0.492i)4-s + (0.667 + 0.242i)5-s + (−0.143 + 0.814i)7-s + (−0.176 + 0.306i)8-s + (0.251 + 0.435i)10-s + (0.627 − 0.228i)11-s + (−0.712 + 0.598i)13-s + (−0.448 + 0.376i)14-s + (−0.234 + 0.0855i)16-s + (0.661 + 1.14i)17-s + (0.158 − 0.275i)19-s + (−0.0616 + 0.349i)20-s + (0.443 + 0.161i)22-s + (0.105 + 0.597i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0931 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0931 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.60322 + 1.76021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60322 + 1.76021i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + (-7.46 - 2.71i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (2.66 - 15.0i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-22.9 + 8.33i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (33.4 - 28.0i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-46.3 - 80.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-13.1 + 22.7i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-11.6 - 65.9i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-74.0 - 62.1i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-44.3 - 251. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (216. + 375. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-299. + 251. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-27.3 + 9.97i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-91.0 + 516. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 541.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-559. - 203. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-47.5 + 269. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-560. + 470. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (53.3 + 92.3i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-497. + 862. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-288. - 242. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (352. + 295. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-320. + 555. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.44e3 - 527. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
| 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
−12.55544547266133625842607134060, −12.04880115045867301860570036467, −10.69569852044137851241060940096, −9.481034684014393959234952224127, −8.584183990457425566711019610537, −7.14990979626969171853389363946, −6.13347709038056815776618950575, −5.23523606021417302727070591561, −3.63687849090952100923319109675, −2.09173821349050853715792558129,
1.00729513338093581183108268748, 2.74682753175763724685328321663, 4.25768864566312712057945924972, 5.40565816907150845740508075381, 6.64444672785304877535002226557, 7.85181248136079522174011755633, 9.608769640734191927506459074986, 9.946404358760585169816853821566, 11.26094661854832734373602334360, 12.22202836265519340014820326257
