L(s) = 1 | − 4i·5-s + 92i·13-s − 94·17-s + 109·25-s − 284i·29-s − 396i·37-s + 230·41-s − 343·49-s + 572i·53-s − 468i·61-s + 368·65-s − 1.09e3·73-s + 376i·85-s − 1.67e3·89-s − 594·97-s + ⋯ |
L(s) = 1 | − 0.357i·5-s + 1.96i·13-s − 1.34·17-s + 0.871·25-s − 1.81i·29-s − 1.75i·37-s + 0.876·41-s − 49-s + 1.48i·53-s − 0.982i·61-s + 0.702·65-s − 1.76·73-s + 0.479i·85-s − 1.98·89-s − 0.621·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6566325527\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6566325527\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4iT - 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 - 92iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 94T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 284iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 396iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 230T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 572iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5T^{2} \) |
| 61 | \( 1 + 468iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.67e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 594T + 9.12e5T^{2} \) |
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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
−9.145224804552795251189529639975, −8.475707962570768734630610448643, −7.36071712365127121825596315504, −6.63330759005090478821176952031, −5.81449674838299737723670751083, −4.47175676802858234722726450670, −4.19187371631371860693197531896, −2.57686930541565764485128091400, −1.65348811902961442924815887329, −0.16134030992495487365135090624,
1.18098686291335506144820966956, 2.66915897433095959557832077903, 3.34910024260917695180001711153, 4.67019392969899697858875913045, 5.44298612168964414983687634898, 6.47652226528099738896484356665, 7.18335257154799982672761882074, 8.186543443507164219810449822741, 8.770756777207221019181577247295, 9.846721667013930985143600456161