L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 4·7-s − 2.82i·8-s − 16.9i·11-s − 8·13-s + 5.65i·14-s + 4.00·16-s − 12.7i·17-s − 16·19-s + 24·22-s − 16.9i·23-s − 11.3i·26-s − 8.00·28-s − 4.24i·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.571·7-s − 0.353i·8-s − 1.54i·11-s − 0.615·13-s + 0.404i·14-s + 0.250·16-s − 0.748i·17-s − 0.842·19-s + 1.09·22-s − 0.737i·23-s − 0.435i·26-s − 0.285·28-s − 0.146i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.369507898\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.369507898\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4T + 49T^{2} \) |
| 11 | \( 1 + 16.9iT - 121T^{2} \) |
| 13 | \( 1 + 8T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 - 44T + 961T^{2} \) |
| 37 | \( 1 - 34T + 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16T + 5.32e3T^{2} \) |
| 79 | \( 1 + 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 176T + 9.40e3T^{2} \) |
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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
−10.80594052948724572388445384316, −9.790436656198777203937546198644, −8.660324312559166151955503080685, −8.180675030583159019148595418883, −7.06028153938759640360044116695, −6.10327590941076757315290499347, −5.15798651477903772444393608744, −4.11438724139480995497635965217, −2.63725051020435494464446810732, −0.59651829725488660624628487980,
1.52612558760377130127707855492, 2.62028100789085471413374289907, 4.23323796104088499016819604541, 4.83899316457483005950746687490, 6.23586915254361910640044429833, 7.47979486438902504691639005918, 8.260640071930376535861670118409, 9.460169139486811842471359441676, 10.06068215057095015217953094954, 10.98629889995339908300020846534