L(s) = 1 | − 5.56i·2-s − 5.19i·3-s − 15.0·4-s − 37.5i·5-s − 28.9·6-s + 91.3i·7-s − 5.54i·8-s − 27·9-s − 208.·10-s + 134. i·11-s + 77.9i·12-s + 280. i·13-s + 508.·14-s − 194.·15-s − 270.·16-s + 68.0·17-s + ⋯ |
L(s) = 1 | − 1.39i·2-s − 0.577i·3-s − 0.937·4-s − 1.50i·5-s − 0.803·6-s + 1.86i·7-s − 0.0866i·8-s − 0.333·9-s − 2.08·10-s + 1.10i·11-s + 0.541i·12-s + 1.66i·13-s + 2.59·14-s − 0.866·15-s − 1.05·16-s + 0.235·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4216089232\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4216089232\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19iT \) |
| 67 | \( 1 + (-1.87e3 - 4.07e3i)T \) |
good | 2 | \( 1 + 5.56iT - 16T^{2} \) |
| 5 | \( 1 + 37.5iT - 625T^{2} \) |
| 7 | \( 1 - 91.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 134. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 280. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 68.0T + 8.35e4T^{2} \) |
| 19 | \( 1 + 402.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 614.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 861.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 485. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 608.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.94e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 789. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.77e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.28e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.16e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.82e3iT - 1.38e7T^{2} \) |
| 71 | \( 1 + 2.07e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 9.34e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.63e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 7.67e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.93e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 441. iT - 8.85e7T^{2} \) |
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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
−12.13870829923112747007385436037, −11.33885618174359596092928777581, −9.639488811119361495440920229722, −9.143055856672041342578807637639, −8.312949856476305108918222822024, −6.55627566277175744220946158339, −5.20395406096576107401846144893, −4.12465969872749596965701144093, −2.08548021587061852182799493917, −1.81604679223602963740283015183,
0.13844994263227391295044718047, 3.09529454987821633501485315834, 4.16791265309237345211312057013, 5.78693453833987800416108269568, 6.53137652556175542302418045973, 7.62950577303141935869801050629, 8.114684311770490802093009736781, 9.904132212222727280062757758959, 10.72757032707186178961544375348, 11.13640849539802183654968051827