L(s) = 1 | + (0.767 − 1.18i)2-s + (−0.822 − 1.82i)4-s − 0.841i·5-s + 2.64·7-s + (−2.79 − 0.420i)8-s + (−0.999 − 0.645i)10-s − 3.91i·11-s + 0.645i·13-s + (2.02 − 3.14i)14-s + (−2.64 + 2.99i)16-s − 5.59·17-s + 4.29i·19-s + (−1.53 + 0.692i)20-s + (−4.64 − 3.00i)22-s + 8.66·23-s + ⋯ |
L(s) = 1 | + (0.542 − 0.840i)2-s + (−0.411 − 0.911i)4-s − 0.376i·5-s + 0.999·7-s + (−0.988 − 0.148i)8-s + (−0.316 − 0.204i)10-s − 1.17i·11-s + 0.179i·13-s + (0.542 − 0.840i)14-s + (−0.661 + 0.749i)16-s − 1.35·17-s + 0.984i·19-s + (−0.343 + 0.154i)20-s + (−0.990 − 0.639i)22-s + 1.80·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02428 - 1.18993i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02428 - 1.18993i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.767 + 1.18i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.841iT - 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 + 3.91iT - 11T^{2} \) |
| 13 | \( 1 - 0.645iT - 13T^{2} \) |
| 17 | \( 1 + 5.59T + 17T^{2} \) |
| 19 | \( 1 - 4.29iT - 19T^{2} \) |
| 23 | \( 1 - 8.66T + 23T^{2} \) |
| 29 | \( 1 - 7.82iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6.64iT - 37T^{2} \) |
| 41 | \( 1 + 6.13T + 41T^{2} \) |
| 43 | \( 1 + 7.29iT - 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 7.82iT - 53T^{2} \) |
| 59 | \( 1 - 7.27iT - 59T^{2} \) |
| 61 | \( 1 + 13.9iT - 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 + 9.50iT - 83T^{2} \) |
| 89 | \( 1 - 5.59T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−11.97773715415822483293696563592, −11.09848483878279041696049766329, −10.57105674362845392084205540006, −8.993338928106170271508068784763, −8.503398962910524119355428671825, −6.74030515119704575054822205626, −5.38585485837981594725335896407, −4.55771632226627770947947903093, −3.11128409266332721575431850417, −1.39264991324237704461846571164,
2.57940208782148185530571697162, 4.39988119399264794708372031902, 5.06980373568661328811280952404, 6.64395192333326584916404006561, 7.30279820599429863364463296021, 8.439809901233853388099151819369, 9.383387451416342370024617698544, 10.89576110249425903270943947476, 11.66506679407841556867920647177, 12.87811252458209336944219133198