L(s) = 1 | − i·2-s − 4-s + 5-s + i·8-s − i·10-s + 5.37i·11-s − 1.19i·13-s + 16-s − 2.17·17-s − 2.71i·19-s − 20-s + 5.37·22-s − 0.496i·23-s + 25-s − 1.19·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447·5-s + 0.353i·8-s − 0.316i·10-s + 1.62i·11-s − 0.332i·13-s + 0.250·16-s − 0.528·17-s − 0.622i·19-s − 0.223·20-s + 1.14·22-s − 0.103i·23-s + 0.200·25-s − 0.235·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.795159727\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.795159727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.37iT - 11T^{2} \) |
| 13 | \( 1 + 1.19iT - 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 2.71iT - 19T^{2} \) |
| 23 | \( 1 + 0.496iT - 23T^{2} \) |
| 29 | \( 1 + 7.12iT - 29T^{2} \) |
| 31 | \( 1 + 0.550iT - 31T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 - 7.42T + 47T^{2} \) |
| 53 | \( 1 - 5.13iT - 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 1.88iT - 61T^{2} \) |
| 67 | \( 1 - 4.25T + 67T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 7.65iT - 97T^{2} \) |
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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
−8.533212665201542229185299110761, −7.55371740783961869705135143515, −6.99612612351872985524284088101, −6.06617740390252257496222494863, −5.21646711648749300526365329659, −4.49015589610912503904350666004, −3.84664098583427781541265032642, −2.50103823453450004411243309506, −2.18771997443452214850038639488, −0.896272727382262917578629968620,
0.63971385935018173210675191552, 1.88776649621781265367959235602, 3.13343470253420420776354516719, 3.81980513491380372759832430950, 4.86675615731583005428263529202, 5.60114154276217326462122660139, 6.15889337101213470982503700580, 6.83754124161448984379664313949, 7.62813431596356670074082250603, 8.485560832033005538282525925476