L(s) = 1 | − 1.79·3-s − 0.791i·7-s + 0.208·9-s + i·11-s + (2 + 3i)13-s + 1.58·17-s − 2.20i·19-s + 1.41i·21-s − 0.791·23-s + 5·25-s + 5.00·27-s + 7.58·29-s + 7.58i·31-s − 1.79i·33-s + 9.16i·37-s + ⋯ |
L(s) = 1 | − 1.03·3-s − 0.299i·7-s + 0.0695·9-s + 0.301i·11-s + (0.554 + 0.832i)13-s + 0.383·17-s − 0.506i·19-s + 0.309i·21-s − 0.164·23-s + 25-s + 0.962·27-s + 1.40·29-s + 1.36i·31-s − 0.311i·33-s + 1.50i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.924279 + 0.279849i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.924279 + 0.279849i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-2 - 3i)T \) |
good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 0.791iT - 7T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 2.20iT - 19T^{2} \) |
| 23 | \( 1 + 0.791T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 7.58iT - 31T^{2} \) |
| 37 | \( 1 - 9.16iT - 37T^{2} \) |
| 41 | \( 1 - 8.37iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 3.16iT - 67T^{2} \) |
| 71 | \( 1 + 16.7iT - 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2.37iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 1.58iT - 97T^{2} \) |
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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
−10.78319220779774881724594478664, −10.24317186375193539668266530629, −9.061284641770976030373396109842, −8.216864752733923578527892972485, −6.85091044883704449160504163225, −6.42974164434514835975167018587, −5.19787037348832356140469889176, −4.46526305184078706919402083200, −2.99931501140151842394212994919, −1.15652894430649401300987467949,
0.78377456461809750835298080707, 2.72382672766988090132954596538, 4.07294661220814787838947456607, 5.43523409944931277757165044538, 5.82797516514771983375067942806, 6.86314243985711874270073507027, 8.049870545382711513519517572529, 8.822956972368713067957114405394, 10.02576625425437365020832789173, 10.78037566085669688908087018597