L(s) = 1 | − 2.91i·2-s + 1.73·3-s − 4.51·4-s − 5.05i·6-s + (3.36 − 6.13i)7-s + 1.49i·8-s + 2.99·9-s − 2.58·11-s − 7.81·12-s + 0.0498·13-s + (−17.9 − 9.80i)14-s − 13.6·16-s + 14.2·17-s − 8.75i·18-s − 14.9i·19-s + ⋯ |
L(s) = 1 | − 1.45i·2-s + 0.577·3-s − 1.12·4-s − 0.842i·6-s + (0.480 − 0.877i)7-s + 0.186i·8-s + 0.333·9-s − 0.235·11-s − 0.651·12-s + 0.00383·13-s + (−1.27 − 0.700i)14-s − 0.855·16-s + 0.835·17-s − 0.486i·18-s − 0.785i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0373i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.000727020\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.000727020\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-3.36 + 6.13i)T \) |
good | 2 | \( 1 + 2.91iT - 4T^{2} \) |
| 11 | \( 1 + 2.58T + 121T^{2} \) |
| 13 | \( 1 - 0.0498T + 169T^{2} \) |
| 17 | \( 1 - 14.2T + 289T^{2} \) |
| 19 | \( 1 + 14.9iT - 361T^{2} \) |
| 23 | \( 1 + 22.2iT - 529T^{2} \) |
| 29 | \( 1 + 17.4T + 841T^{2} \) |
| 31 | \( 1 + 6.36iT - 961T^{2} \) |
| 37 | \( 1 - 7.14iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 74.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 79.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 81.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 67.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 4.33iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 109. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 49.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 97.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 116.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 98.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 112.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 77.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 109.T + 9.40e3T^{2} \) |
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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
−10.29288384662963179846631822956, −9.655816549245952543032573958040, −8.644543634991386195058420406935, −7.69349166408545608199073402796, −6.70836773997521260679183886086, −4.99417515534535445801307996894, −4.04769617154673325509731296345, −3.12240406804953722201009744008, −1.99187421001700082476843836839, −0.74308389930456965461877752565,
1.92025134452261917934514432415, 3.42664059541643540354291785770, 4.91059180650183553286903983789, 5.61954380263356482836615245059, 6.55786901206730958054241682046, 7.78842842999993230573384776986, 8.026253515250147579493652070633, 9.049276955040283751180687698602, 9.779722518916714051381258954085, 11.10958036556255609033071259655