L(s) = 1 | + (−1.21 + 0.723i)2-s + 3-s + (0.952 − 1.75i)4-s − 4.31·5-s + (−1.21 + 0.723i)6-s + 1.64·7-s + (0.114 + 2.82i)8-s + 9-s + (5.23 − 3.11i)10-s − 2.66i·11-s + (0.952 − 1.75i)12-s + 4.33i·13-s + (−2.00 + 1.19i)14-s − 4.31·15-s + (−2.18 − 3.35i)16-s − 0.417i·17-s + ⋯ |
L(s) = 1 | + (−0.859 + 0.511i)2-s + 0.577·3-s + (0.476 − 0.879i)4-s − 1.92·5-s + (−0.496 + 0.295i)6-s + 0.622·7-s + (0.0406 + 0.999i)8-s + 0.333·9-s + (1.65 − 0.986i)10-s − 0.803i·11-s + (0.275 − 0.507i)12-s + 1.20i·13-s + (−0.534 + 0.318i)14-s − 1.11·15-s + (−0.546 − 0.837i)16-s − 0.101i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.690692 - 0.297057i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.690692 - 0.297057i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.21 - 0.723i)T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + (-3.61 + 3.15i)T \) |
good | 5 | \( 1 + 4.31T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 2.66iT - 11T^{2} \) |
| 13 | \( 1 - 4.33iT - 13T^{2} \) |
| 17 | \( 1 + 0.417iT - 17T^{2} \) |
| 19 | \( 1 + 7.23iT - 19T^{2} \) |
| 29 | \( 1 + 9.62iT - 29T^{2} \) |
| 31 | \( 1 + 6.72iT - 31T^{2} \) |
| 37 | \( 1 + 5.50T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 - 1.68iT - 43T^{2} \) |
| 47 | \( 1 + 1.93iT - 47T^{2} \) |
| 53 | \( 1 - 5.44T + 53T^{2} \) |
| 59 | \( 1 - 9.10T + 59T^{2} \) |
| 61 | \( 1 + 0.402T + 61T^{2} \) |
| 67 | \( 1 + 5.42iT - 67T^{2} \) |
| 71 | \( 1 + 1.27iT - 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 11.0iT - 83T^{2} \) |
| 89 | \( 1 - 18.4iT - 89T^{2} \) |
| 97 | \( 1 + 9.76iT - 97T^{2} \) |
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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
−10.87314058527969178677913553027, −9.416802513506525458207846584312, −8.647523852900237073392515293256, −8.124733374565541982171014381577, −7.31241427700734566817126755489, −6.60532050371825357024103919526, −4.89464362056362025848846451160, −4.04493245941247201178130216323, −2.56452665265709549143239345698, −0.59924994144787402783459454918,
1.37381904231531750390969000528, 3.15764754231466661490863017129, 3.76887744449877441476196058619, 4.98431034577283716096122203901, 7.09695222380098081379598764865, 7.55571621939565766785332472278, 8.305846740304114382085723871662, 8.825780322034333935683588637110, 10.23004533757685943858319574020, 10.75343289506918265199740303343