| L(s) = 1 | − 9·3-s − 33.0i·5-s + 61.2i·7-s + 81·9-s + 164. i·11-s + (−443. + 417. i)13-s + 297. i·15-s + 336.·17-s + 69.6i·19-s − 551. i·21-s − 3.73e3·23-s + 2.03e3·25-s − 729·27-s + 4.30e3·29-s − 1.85e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.591i·5-s + 0.472i·7-s + 0.333·9-s + 0.410i·11-s + (−0.727 + 0.685i)13-s + 0.341i·15-s + 0.282·17-s + 0.0442i·19-s − 0.272i·21-s − 1.47·23-s + 0.650·25-s − 0.192·27-s + 0.950·29-s − 0.345i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4047184168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4047184168\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 13 | \( 1 + (443. - 417. i)T \) |
| good | 5 | \( 1 + 33.0iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 61.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 164. iT - 1.61e5T^{2} \) |
| 17 | \( 1 - 336.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 69.6iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.85e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 2.73e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.20e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.16e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.03e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 696.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.42e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.86e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.63e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 8.37e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.85e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.40e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 8.73e4iT - 8.58e9T^{2} \) |
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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
−9.594875405672921215233662873505, −8.685462944364195510134370858895, −7.73600939199366920040994848041, −6.73548468769692341503113403145, −5.81543646847883110422149216338, −4.89284365950255543310569194522, −4.13507870525019502984986974555, −2.55035684222680872200842907481, −1.42824408489393974091797167423, −0.11036701592707312095807058524,
0.987217426542848906393094559167, 2.48837013734943451908246258424, 3.58121956570458921271735987118, 4.70802418925461832163991844027, 5.71133189284422302858144532982, 6.58480493649862435851448345709, 7.43896015910447111306512006728, 8.255139212061635515078059450482, 9.508707851383379610988086244090, 10.47030981568512472115884419266
