| L(s) = 1 | − 0.153i·3-s − 2.94i·5-s + (0.319 + 2.62i)7-s + 2.97·9-s + (1.01 + 3.15i)11-s − 13-s − 0.452·15-s + 5.02·17-s + 4.07·19-s + (0.403 − 0.0490i)21-s − 6.95·23-s − 3.69·25-s − 0.917i·27-s + 5.04i·29-s − 0.575i·31-s + ⋯ |
| L(s) = 1 | − 0.0886i·3-s − 1.31i·5-s + (0.120 + 0.992i)7-s + 0.992·9-s + (0.306 + 0.951i)11-s − 0.277·13-s − 0.116·15-s + 1.21·17-s + 0.935·19-s + (0.0879 − 0.0107i)21-s − 1.45·23-s − 0.739·25-s − 0.176i·27-s + 0.936i·29-s − 0.103i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.149463997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149463997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.319 - 2.62i)T \) |
| 11 | \( 1 + (-1.01 - 3.15i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.153iT - 3T^{2} \) |
| 5 | \( 1 + 2.94iT - 5T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 - 4.07T + 19T^{2} \) |
| 23 | \( 1 + 6.95T + 23T^{2} \) |
| 29 | \( 1 - 5.04iT - 29T^{2} \) |
| 31 | \( 1 + 0.575iT - 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 - 5.12iT - 43T^{2} \) |
| 47 | \( 1 - 6.71iT - 47T^{2} \) |
| 53 | \( 1 + 1.82T + 53T^{2} \) |
| 59 | \( 1 - 13.5iT - 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 0.423T + 73T^{2} \) |
| 79 | \( 1 - 8.71iT - 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 + 4.93iT - 89T^{2} \) |
| 97 | \( 1 - 0.793iT - 97T^{2} \) |
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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
−8.484182877770742968304021824136, −7.76407508760190956775804372610, −7.26751403316094255161632737191, −6.09032847787431600777221790521, −5.48739129638872827715104202654, −4.69313185123313704826582507792, −4.19811769868418736090694043182, −2.95596621562629160644189897972, −1.76241880162342491857377847904, −1.15752025238813899975728594338,
0.70767018253645977588760301893, 1.89000515791550366639579376114, 3.14349090052851226288921060816, 3.65837979281497524316909655890, 4.41218375651125178934985371592, 5.56090234087103249353966401306, 6.31132135528321599797455054653, 6.98942929248234386767279966268, 7.67416905195310719879513605785, 8.015447387062611270511758444425
