| L(s) = 1 | + 1.60i·5-s + 7-s + 0.0549i·11-s − 3.75i·13-s − 3.16·17-s + 0.726i·19-s − 7.77·23-s + 2.41·25-s − 2.75i·29-s + 2.14·31-s + 1.60i·35-s − 0.600i·37-s + 4.53·41-s + 6.85i·43-s − 0.744·47-s + ⋯ |
| L(s) = 1 | + 0.718i·5-s + 0.377·7-s + 0.0165i·11-s − 1.04i·13-s − 0.766·17-s + 0.166i·19-s − 1.62·23-s + 0.483·25-s − 0.512i·29-s + 0.385·31-s + 0.271i·35-s − 0.0987i·37-s + 0.708·41-s + 1.04i·43-s − 0.108·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.066872941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066872941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 1.60iT - 5T^{2} \) |
| 11 | \( 1 - 0.0549iT - 11T^{2} \) |
| 13 | \( 1 + 3.75iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.726iT - 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 + 2.75iT - 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 0.600iT - 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 - 6.85iT - 43T^{2} \) |
| 47 | \( 1 + 0.744T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 2.58iT - 59T^{2} \) |
| 61 | \( 1 - 9.80iT - 61T^{2} \) |
| 67 | \( 1 - 12.2iT - 67T^{2} \) |
| 71 | \( 1 + 9.19T + 71T^{2} \) |
| 73 | \( 1 - 3.85T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 7.90T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.232317179632578937364679836209, −7.62604984405640838111369823162, −6.98160224135288284676494838869, −6.07395993559625087482062797194, −5.69419671781483677666026789029, −4.57328462841213539274018688763, −4.00585925864239316933270383596, −2.92189445262588551469934960940, −2.38240937646334405216427050374, −1.15479111511585884856525419584,
0.27540053526496939001118057155, 1.59857031474942986376177477844, 2.22914019484091124148329022816, 3.47045860698091191302265463758, 4.36400844002157529139168787413, 4.76894645765294032805142735000, 5.65485544556012692624732054599, 6.45418216119642479149176371695, 7.06925220576501175910738691067, 7.967346471280940399344874008785
