L(s) = 1 | + (−0.207 + 1.39i)2-s + (−1.91 − 0.579i)4-s + 2.61i·5-s + (1.20 − 2.55i)8-s + (−3.65 − 0.541i)10-s − 3.95i·11-s − 1.08i·13-s + (3.32 + 2.21i)16-s + 0.317i·17-s + 5.16·19-s + (1.51 − 5.00i)20-s + (5.53 + 0.819i)22-s − 2.31i·23-s − 1.82·25-s + (1.51 + 0.224i)26-s + ⋯ |
L(s) = 1 | + (−0.146 + 0.989i)2-s + (−0.957 − 0.289i)4-s + 1.16i·5-s + (0.426 − 0.904i)8-s + (−1.15 − 0.171i)10-s − 1.19i·11-s − 0.300i·13-s + (0.832 + 0.554i)16-s + 0.0768i·17-s + 1.18·19-s + (0.338 − 1.11i)20-s + (1.18 + 0.174i)22-s − 0.483i·23-s − 0.365·25-s + (0.296 + 0.0439i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.463660963\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.463660963\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.207 - 1.39i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.61iT - 5T^{2} \) |
| 11 | \( 1 + 3.95iT - 11T^{2} \) |
| 13 | \( 1 + 1.08iT - 13T^{2} \) |
| 17 | \( 1 - 0.317iT - 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 + 2.31iT - 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2.29iT - 41T^{2} \) |
| 43 | \( 1 - 7.23iT - 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 5.41iT - 61T^{2} \) |
| 67 | \( 1 + 3.27iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 14.0iT - 73T^{2} \) |
| 79 | \( 1 + 7.91iT - 79T^{2} \) |
| 83 | \( 1 - 9.45T + 83T^{2} \) |
| 89 | \( 1 - 5.99iT - 89T^{2} \) |
| 97 | \( 1 + 9.23iT - 97T^{2} \) |
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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
−9.400340620675095797645449840446, −8.418808961827942545368165151014, −7.928404807665068708982625000493, −6.95245121644489598522776459289, −6.42553959191129987387624299685, −5.67298602295219792303550127849, −4.75071825951534237942735675401, −3.53646425887043738388242517555, −2.83113023157235481318387535669, −0.877742154666773418036744179240,
0.892584406168074569971257805463, 1.84101042599861127917761655037, 3.01304089291078148212495002278, 4.18616032141691766622889507368, 4.81924773519571367647847239303, 5.46235934139367838340047991359, 6.87793438835241409540240290939, 7.81285109768696779422279463136, 8.547968895015527867687038004489, 9.283171920457365678085571472564