| L(s) = 1 | + 1.59i·3-s + (−1.59 − 1.56i)5-s + 1.66i·7-s + 0.463·9-s + 5.56·11-s − 6.31i·13-s + (2.48 − 2.54i)15-s + 4.12i·17-s − 19-s − 2.64·21-s + 1.82i·23-s + (0.114 + 4.99i)25-s + 5.51i·27-s + 4.08·29-s + 6.61·31-s + ⋯ |
| L(s) = 1 | + 0.919i·3-s + (−0.715 − 0.698i)5-s + 0.628i·7-s + 0.154·9-s + 1.67·11-s − 1.75i·13-s + (0.642 − 0.657i)15-s + 0.999i·17-s − 0.229·19-s − 0.577·21-s + 0.380i·23-s + (0.0228 + 0.999i)25-s + 1.06i·27-s + 0.759·29-s + 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40462 + 0.591230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40462 + 0.591230i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.59 + 1.56i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.59iT - 3T^{2} \) |
| 7 | \( 1 - 1.66iT - 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 6.31iT - 13T^{2} \) |
| 17 | \( 1 - 4.12iT - 17T^{2} \) |
| 23 | \( 1 - 1.82iT - 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 - 9.66iT - 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 - 3.75iT - 43T^{2} \) |
| 47 | \( 1 + 3.85iT - 47T^{2} \) |
| 53 | \( 1 + 5.24iT - 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 0.155iT - 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 0.795iT - 73T^{2} \) |
| 79 | \( 1 - 7.18T + 79T^{2} \) |
| 83 | \( 1 + 5.88iT - 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 2.77iT - 97T^{2} \) |
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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
−10.22892243132131939814432430363, −9.720932829680840335506758582726, −8.529180525645778800779709902434, −8.360445848053891872653996964625, −6.93908069361247496266239138460, −5.84610157080938846926080604428, −4.88249879759487678289728892353, −4.04116832240462592217714432245, −3.22714023704461296145597557069, −1.23010730108682489659018207044,
1.01744621884474014362479267323, 2.34828176369544434625206606311, 3.91601094573919204294430901812, 4.40408729270776271137497397592, 6.33618770564950780982799802877, 6.93300309704741332280698591749, 7.21694408319060245980913450831, 8.434641436070484929294332894114, 9.316965018973412042982228443894, 10.26537963225840132677524599825
