L(s) = 1 | + i·5-s + 1.56i·7-s + 0.438i·11-s + (0.561 + 3.56i)13-s − 6.68·17-s − 5.56·23-s − 25-s − 2·29-s − 3.12i·31-s − 1.56·35-s − 5.56i·37-s − 9.80i·41-s − 3.12·43-s + 9.12i·47-s + 4.56·49-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.590i·7-s + 0.132i·11-s + (0.155 + 0.987i)13-s − 1.62·17-s − 1.15·23-s − 0.200·25-s − 0.371·29-s − 0.560i·31-s − 0.263·35-s − 0.914i·37-s − 1.53i·41-s − 0.476·43-s + 1.33i·47-s + 0.651·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3707469864\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3707469864\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-0.561 - 3.56i)T \) |
good | 7 | \( 1 - 1.56iT - 7T^{2} \) |
| 11 | \( 1 - 0.438iT - 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.12iT - 31T^{2} \) |
| 37 | \( 1 + 5.56iT - 37T^{2} \) |
| 41 | \( 1 + 9.80iT - 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 - 9.12iT - 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 + 1.12iT - 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 2.24iT - 67T^{2} \) |
| 71 | \( 1 + 0.438iT - 71T^{2} \) |
| 73 | \( 1 - 7.12iT - 73T^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 4.43iT - 89T^{2} \) |
| 97 | \( 1 + 9.56iT - 97T^{2} \) |
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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
−9.166446486430118427709776245246, −8.858118265302254606135128672122, −7.81259856406914301020444907452, −7.02096113756442423983750979041, −6.30748630478917155841175629634, −5.61857716408052131827657946407, −4.45416668643632719110419776791, −3.85060962819057796519908436484, −2.50997343276166515918927522977, −1.89269766778583829164182331200,
0.11918976136865344293876148233, 1.46143341521250491204987042127, 2.67796565269779551255403486109, 3.77858985987400703570159886358, 4.52867534458957099780907753997, 5.38492497452504313688416133251, 6.29230840261715827968408936683, 7.00908960211410835299290730890, 7.989834783101050853710053387407, 8.463575876130162193718213140720