| L(s) = 1 | + (−0.718 − 1.21i)2-s + (1.70 − 0.298i)3-s + (−0.967 + 1.75i)4-s + (−1.03 + 2.85i)5-s + (−1.58 − 1.86i)6-s + (−0.984 − 0.173i)7-s + (2.82 − 0.0783i)8-s + (2.82 − 1.01i)9-s + (4.22 − 0.784i)10-s + (−1.50 + 0.546i)11-s + (−1.12 + 3.27i)12-s + (−2.67 + 2.24i)13-s + (0.495 + 1.32i)14-s + (−0.920 + 5.17i)15-s + (−2.12 − 3.38i)16-s + (−1.75 + 1.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.507 − 0.861i)2-s + (0.985 − 0.172i)3-s + (−0.483 + 0.875i)4-s + (−0.464 + 1.27i)5-s + (−0.648 − 0.761i)6-s + (−0.372 − 0.0656i)7-s + (0.999 − 0.0277i)8-s + (0.940 − 0.339i)9-s + (1.33 − 0.248i)10-s + (−0.452 + 0.164i)11-s + (−0.326 + 0.945i)12-s + (−0.742 + 0.623i)13-s + (0.132 + 0.353i)14-s + (−0.237 + 1.33i)15-s + (−0.531 − 0.846i)16-s + (−0.425 + 0.245i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.828850 + 0.553435i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.828850 + 0.553435i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.718 + 1.21i)T \) |
| 3 | \( 1 + (-1.70 + 0.298i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (1.03 - 2.85i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (1.50 - 0.546i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.67 - 2.24i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.75 - 1.01i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.389 - 0.224i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.640 - 3.63i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.75 - 6.86i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.43 - 0.253i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.02 + 3.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.73 - 5.64i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.51 - 9.66i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.381 - 2.16i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (4.14 + 1.50i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.36 + 7.72i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.90 + 10.6i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.14 + 5.43i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.22 + 10.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.61 + 1.93i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.37 - 5.34i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-13.8 - 8.01i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.00 + 2.54i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.61747088650290936365190615273, −9.468467344148386643896513974563, −9.179904051210189054987468228111, −7.69915834489973509217151937452, −7.52876643022761930984513800798, −6.53646927432202667042931127894, −4.61689579633596052835251033606, −3.53074976741463876006357129872, −2.90895921255307313964546743433, −1.87345400866854461175546656664,
0.52025235369721862779737226749, 2.28742718211935787869287768872, 3.92979753556178461756360124483, 4.81389166731849364809043298250, 5.64289559668638400103424817279, 7.07783231542167142881844557397, 7.75658401804552262566168944291, 8.531678983055889861366793466212, 9.038129778803831881198696435530, 9.847450291358851627936634695360
