| L(s) = 1 | + 4.24·5-s + 8.48·7-s + 4·11-s + 18i·13-s + 4.24i·17-s + 16.9i·19-s + 36i·23-s − 7.00·25-s + 12.7·29-s − 8.48·31-s + 35.9·35-s − 36i·37-s − 29.6i·41-s + 67.8i·43-s − 36i·47-s + ⋯ |
| L(s) = 1 | + 0.848·5-s + 1.21·7-s + 0.363·11-s + 1.38i·13-s + 0.249i·17-s + 0.893i·19-s + 1.56i·23-s − 0.280·25-s + 0.438·29-s − 0.273·31-s + 1.02·35-s − 0.972i·37-s − 0.724i·41-s + 1.57i·43-s − 0.765i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.605611922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.605611922\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.24T + 25T^{2} \) |
| 7 | \( 1 - 8.48T + 49T^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 - 18iT - 169T^{2} \) |
| 17 | \( 1 - 4.24iT - 289T^{2} \) |
| 19 | \( 1 - 16.9iT - 361T^{2} \) |
| 23 | \( 1 - 36iT - 529T^{2} \) |
| 29 | \( 1 - 12.7T + 841T^{2} \) |
| 31 | \( 1 + 8.48T + 961T^{2} \) |
| 37 | \( 1 + 36iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 29.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 67.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 36iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 80.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 80T + 3.48e3T^{2} \) |
| 61 | \( 1 + 36iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 118. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 108iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 56T + 5.32e3T^{2} \) |
| 79 | \( 1 + 25.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 76T + 6.88e3T^{2} \) |
| 89 | \( 1 + 89.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 104T + 9.40e3T^{2} \) |
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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
−9.621254388192596983554375211011, −9.042232329244897578602523986411, −8.090281232543421000289904416148, −7.30352186409615845670280283930, −6.27081626659320399115143260997, −5.52623766330207826741934795075, −4.58647272851736845363693396776, −3.65203925398221482751252996849, −1.99443565508377841565234601681, −1.51569717919210617641285910935,
0.803803015436518391410579244605, 2.02300587389276465915435975041, 3.01603773321547492804364745141, 4.49258734848862739914239470963, 5.16312554061074105218085385110, 6.03715155375870151585396668912, 6.95571019548950905130633028107, 8.014719676207113382511822699986, 8.553408990268355523318219562029, 9.521795636956124574908103418003
