| L(s) = 1 | + 9.83·5-s − 13.5·7-s − 40.0·11-s − 46.4·13-s − 42.4·17-s + 39.9·19-s + 38.8·23-s − 28.2·25-s − 155.·29-s + 132.·31-s − 133.·35-s − 116.·37-s + 7.90·41-s + 455.·43-s + 270.·47-s − 159.·49-s − 71.3·53-s − 394.·55-s + 779.·59-s + 790.·61-s − 457.·65-s + 835.·67-s − 249.·71-s + 619.·73-s + 542.·77-s − 262.·79-s + 465.·83-s + ⋯ |
| L(s) = 1 | + 0.879·5-s − 0.731·7-s − 1.09·11-s − 0.991·13-s − 0.605·17-s + 0.482·19-s + 0.351·23-s − 0.225·25-s − 0.992·29-s + 0.769·31-s − 0.643·35-s − 0.519·37-s + 0.0301·41-s + 1.61·43-s + 0.840·47-s − 0.465·49-s − 0.184·53-s − 0.965·55-s + 1.71·59-s + 1.66·61-s − 0.872·65-s + 1.52·67-s − 0.416·71-s + 0.992·73-s + 0.802·77-s − 0.373·79-s + 0.615·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.634876531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634876531\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 9.83T + 125T^{2} \) |
| 7 | \( 1 + 13.5T + 343T^{2} \) |
| 11 | \( 1 + 40.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 155.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 116.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 7.90T + 6.89e4T^{2} \) |
| 43 | \( 1 - 455.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 270.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 71.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 779.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 790.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 835.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 249.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 262.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 465.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 891.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 248.T + 9.12e5T^{2} \) |
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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
−9.201690781319806136301876175659, −8.140520250768265099657576894465, −7.30005691727236580318354144924, −6.58113546116069983445178148871, −5.60267117407354665457517362162, −5.11116359555265983067116384279, −3.89994052294860438691287870905, −2.70784391681495322964279915324, −2.14787760473251135102058855939, −0.57625877096312671032290145955,
0.57625877096312671032290145955, 2.14787760473251135102058855939, 2.70784391681495322964279915324, 3.89994052294860438691287870905, 5.11116359555265983067116384279, 5.60267117407354665457517362162, 6.58113546116069983445178148871, 7.30005691727236580318354144924, 8.140520250768265099657576894465, 9.201690781319806136301876175659
