| 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 | \( 1 + (1.44e3 + 527. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.22202836265519340014820326257, −11.26094661854832734373602334360, −9.946404358760585169816853821566, −9.608769640734191927506459074986, −7.85181248136079522174011755633, −6.64444672785304877535002226557, −5.40565816907150845740508075381, −4.25768864566312712057945924972, −2.74682753175763724685328321663, −1.00729513338093581183108268748,
2.09173821349050853715792558129, 3.63687849090952100923319109675, 5.23523606021417302727070591561, 6.13347709038056815776618950575, 7.14990979626969171853389363946, 8.584183990457425566711019610537, 9.481034684014393959234952224127, 10.69569852044137851241060940096, 12.04880115045867301860570036467, 12.55544547266133625842607134060
