L(s) = 1 | + (1.36 + 0.366i)2-s − 1.73i·3-s + (1.73 + i)4-s + (0.633 − 2.36i)6-s + (2 + 1.73i)7-s + (1.99 + 2i)8-s − 0.267i·11-s + (1.73 − 2.99i)12-s + 0.464·13-s + (2.09 + 3.09i)14-s + (1.99 + 3.46i)16-s − 6.46·17-s + 6·19-s + (2.99 − 3.46i)21-s + (0.0980 − 0.366i)22-s − 1.46·23-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s − 0.999i·3-s + (0.866 + 0.5i)4-s + (0.258 − 0.965i)6-s + (0.755 + 0.654i)7-s + (0.707 + 0.707i)8-s − 0.0807i·11-s + (0.499 − 0.866i)12-s + 0.128·13-s + (0.560 + 0.827i)14-s + (0.499 + 0.866i)16-s − 1.56·17-s + 1.37·19-s + (0.654 − 0.755i)21-s + (0.0209 − 0.0780i)22-s − 0.305·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.02655 - 0.197268i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.02655 - 0.197268i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 11 | \( 1 + 0.267iT - 11T^{2} \) |
| 13 | \( 1 - 0.464T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 9.46iT - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 1.73iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 9.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 7.46iT - 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 - 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.53iT - 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−10.90842979723042272774567100962, −9.441485246980431746671809575868, −8.331897858126787419164375350693, −7.62217591824559592025355855341, −6.85294275364950258443460882618, −5.97429169140150058832547701873, −5.08628884980576123187165154967, −4.03860327509330806945508217136, −2.57404558513679197512543183829, −1.66023683557356916445291105292,
1.58287794900003810288525133734, 3.11848015090940850735292749760, 4.22473694104050886964546344925, 4.65718112286942384447881635206, 5.64308950738007460714633953993, 6.86276988661723710345436337874, 7.64885325032752486539028716986, 8.958580903358736645421288072367, 9.959233435931351821087204634839, 10.52952016666067537165298231931