| L(s) = 1 | + (0.503 + 1.32i)2-s + (−1.58 + 0.707i)3-s + (−1.49 + 1.32i)4-s + (−2.77 + 2.27i)5-s + (−1.73 − 1.73i)6-s + (1.08 + 1.63i)7-s + (−2.50 − 1.30i)8-s + (1.99 − 2.23i)9-s + (−4.39 − 2.51i)10-s + (0.773 − 0.234i)11-s + (1.42 − 3.15i)12-s + (1.37 + 1.13i)13-s + (−1.60 + 2.25i)14-s + (2.76 − 5.55i)15-s + (0.463 − 3.97i)16-s + (−1.91 − 4.61i)17-s + ⋯ |
| L(s) = 1 | + (0.355 + 0.934i)2-s + (−0.912 + 0.408i)3-s + (−0.746 + 0.664i)4-s + (−1.23 + 1.01i)5-s + (−0.706 − 0.707i)6-s + (0.411 + 0.616i)7-s + (−0.887 − 0.461i)8-s + (0.665 − 0.746i)9-s + (−1.39 − 0.796i)10-s + (0.233 − 0.0707i)11-s + (0.410 − 0.912i)12-s + (0.382 + 0.313i)13-s + (−0.429 + 0.603i)14-s + (0.715 − 1.43i)15-s + (0.115 − 0.993i)16-s + (−0.463 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.261508 - 0.363032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.261508 - 0.363032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.503 - 1.32i)T \) |
| 3 | \( 1 + (1.58 - 0.707i)T \) |
| good | 5 | \( 1 + (2.77 - 2.27i)T + (0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.08 - 1.63i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.773 + 0.234i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 1.13i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (1.91 + 4.61i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (4.68 - 0.461i)T + (18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (1.09 - 0.218i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-8.16 - 2.47i)T + (24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (3.89 - 3.89i)T - 31iT^{2} \) |
| 37 | \( 1 + (6.27 + 0.617i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (7.63 - 1.51i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (5.80 - 10.8i)T + (-23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-1.62 - 3.92i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-4.60 + 1.39i)T + (44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-0.748 + 0.614i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (4.17 + 7.81i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (3.05 - 1.63i)T + (37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (1.49 + 2.23i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-8.92 - 5.96i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.22 - 0.920i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (16.0 - 1.58i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-0.636 + 3.19i)T + (-82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-9.83 - 9.83i)T + 97iT^{2} \) |
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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
−11.83285212088443356665516291703, −11.32748097847562301873304090918, −10.31544768412644169143483340986, −8.982317349024904904199511850258, −8.128293747225628820845572015045, −6.88238994445598848700853113174, −6.53720185392707767605118461970, −5.15708326641628512270123517605, −4.29876342231480907151053970938, −3.23232279849836380513591023958,
0.30908745523680396698247205697, 1.64873701676851880930891851553, 3.94228951841838482083255276095, 4.44615857806180821865870058927, 5.51950416208729337564886751336, 6.78539279525184259165384303441, 8.149149842396333024741059091223, 8.697936698316346978707611252508, 10.36959896874107589298287940363, 10.80028929644604745830742523926
