| L(s) = 1 | + 2.23·5-s + 6.33i·7-s − 9.27i·11-s − 18.5·13-s − 13.9·17-s − 17.2i·19-s + 33.7i·23-s + 5.00·25-s − 28.6·29-s − 23.4i·31-s + 14.1i·35-s + 67.3·37-s + 44.0·41-s − 50.2i·43-s + 31.1i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.904i·7-s − 0.843i·11-s − 1.42·13-s − 0.818·17-s − 0.907i·19-s + 1.46i·23-s + 0.200·25-s − 0.986·29-s − 0.757i·31-s + 0.404i·35-s + 1.81·37-s + 1.07·41-s − 1.16i·43-s + 0.662i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.850306049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850306049\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 - 6.33iT - 49T^{2} \) |
| 11 | \( 1 + 9.27iT - 121T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 + 13.9T + 289T^{2} \) |
| 19 | \( 1 + 17.2iT - 361T^{2} \) |
| 23 | \( 1 - 33.7iT - 529T^{2} \) |
| 29 | \( 1 + 28.6T + 841T^{2} \) |
| 31 | \( 1 + 23.4iT - 961T^{2} \) |
| 37 | \( 1 - 67.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 50.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 81.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.49iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 13.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 69.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 46.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 68.5T + 9.40e3T^{2} \) |
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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
−8.825065630692186817641928493703, −7.75745927284599090289599318609, −7.18083522195249947466938600792, −6.13242562789847307138987349830, −5.58450998230677333953511037354, −4.86302405686484954032593582756, −3.81557282376440117403791015509, −2.60655305613767744654625907831, −2.19410306159402027282572790299, −0.62325216423341969149576869392,
0.68265048082619791175785036325, 1.99810081047598688961593552739, 2.69547348008305176985470006788, 4.10924027637101910686060268580, 4.53289603969289996609705466748, 5.47948689510665490116478286158, 6.45168472551920189769769047248, 7.13467932360208280283378014635, 7.67803595097809297434872493664, 8.597004760871339365044369441988
