L(s) = 1 | + 3i·3-s − 8i·5-s − 12·7-s − 9·9-s + 12i·11-s − 20i·13-s + 24·15-s + 62·17-s + 108i·19-s − 36i·21-s + 72·23-s + 61·25-s − 27i·27-s + 128i·29-s + 204·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.715i·5-s − 0.647·7-s − 0.333·9-s + 0.328i·11-s − 0.426i·13-s + 0.413·15-s + 0.884·17-s + 1.30i·19-s − 0.374i·21-s + 0.652·23-s + 0.487·25-s − 0.192i·27-s + 0.819i·29-s + 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.662048924\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.662048924\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
good | 5 | \( 1 + 8iT - 125T^{2} \) |
| 7 | \( 1 + 12T + 343T^{2} \) |
| 11 | \( 1 - 12iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 62T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 204T + 2.97e4T^{2} \) |
| 37 | \( 1 + 228iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 22T + 6.89e4T^{2} \) |
| 43 | \( 1 - 204iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 600T + 1.03e5T^{2} \) |
| 53 | \( 1 - 256iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 828iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 84iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 348iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 456T + 3.57e5T^{2} \) |
| 73 | \( 1 - 822T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.35e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 108iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 938T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.27e3T + 9.12e5T^{2} \) |
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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
−10.80794673550282631160108035630, −10.07581722560803102455752514939, −9.278962047227124056166872836397, −8.405451713842020603646094967365, −7.36200899198744230967914767662, −6.03029862734700909799207496381, −5.16738416771866755775375752886, −4.04303773165259767684355044715, −2.92198359829609070233142221255, −1.04797873822628727908765330350,
0.72028353492479047861093909844, 2.50033972193491986500237740914, 3.41710824678293383153661424220, 4.99321205682029885702958347931, 6.33506803100283482008421661977, 6.84660943388273585944637826379, 7.88366026597079824288528619576, 8.970381761779671909165505777745, 9.908757253579809690057150085985, 10.88758045829862777463725896634