L(s) = 1 | − 8.11·5-s + 9.48i·7-s + 10.5i·11-s + 20.9·13-s + 4.92·17-s − 26.9i·19-s + 43.7i·23-s + 40.9·25-s − 14.5·29-s + 25.4i·31-s − 77.0i·35-s − 12.9·37-s − 50.1·41-s + 5.03i·43-s − 40.8i·47-s + ⋯ |
L(s) = 1 | − 1.62·5-s + 1.35i·7-s + 0.962i·11-s + 1.61·13-s + 0.289·17-s − 1.41i·19-s + 1.90i·23-s + 1.63·25-s − 0.500·29-s + 0.822i·31-s − 2.20i·35-s − 0.350·37-s − 1.22·41-s + 0.117i·43-s − 0.869i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5991628000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5991628000\) |
\(L(2)\) |
|
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 | \( 1 + 8.11T + 25T^{2} \) |
| 7 | \( 1 - 9.48iT - 49T^{2} \) |
| 11 | \( 1 - 10.5iT - 121T^{2} \) |
| 13 | \( 1 - 20.9T + 169T^{2} \) |
| 17 | \( 1 - 4.92T + 289T^{2} \) |
| 19 | \( 1 + 26.9iT - 361T^{2} \) |
| 23 | \( 1 - 43.7iT - 529T^{2} \) |
| 29 | \( 1 + 14.5T + 841T^{2} \) |
| 31 | \( 1 - 25.4iT - 961T^{2} \) |
| 37 | \( 1 + 12.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.03iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 27.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 65.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 127.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 35.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 15.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 73.8T + 9.40e3T^{2} \) |
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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
−9.942101127591931474972937724291, −8.855248768901441756267347723543, −8.575443435518729772097167871399, −7.53182167981129571616677391943, −6.88810486108895329064738224853, −5.69979306965355479750746827746, −4.85414329507171873071366981564, −3.78692396507640589353005898672, −3.03848017486895691339684665345, −1.54115859415924529939334851710,
0.21072101565080779048047083565, 1.19313536600033497415173466280, 3.32936345853287489197619203496, 3.80398957336384929283414269898, 4.50126395345419407297631080966, 5.97486746559267640603804327772, 6.74528847315641836876639571323, 7.84315917704253424621027089046, 8.104789118029954667524865549392, 8.941902948598354778144112722463