| L(s) = 1 | − 1.10i·2-s + 0.786·4-s + 2.47·5-s − 3.06i·8-s − 2.72i·10-s + 3.81i·11-s − 0.163i·13-s − 1.80·16-s + 5.85·17-s − 7.11i·19-s + 1.94·20-s + 4.20·22-s + 5.54i·23-s + 1.13·25-s − 0.180·26-s + ⋯ |
| L(s) = 1 | − 0.779i·2-s + 0.393·4-s + 1.10·5-s − 1.08i·8-s − 0.862i·10-s + 1.14i·11-s − 0.0453i·13-s − 0.452·16-s + 1.41·17-s − 1.63i·19-s + 0.435·20-s + 0.895·22-s + 1.15i·23-s + 0.226·25-s − 0.0353·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.498710778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.498710778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.10iT - 2T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 11 | \( 1 - 3.81iT - 11T^{2} \) |
| 13 | \( 1 + 0.163iT - 13T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 + 7.11iT - 19T^{2} \) |
| 23 | \( 1 - 5.54iT - 23T^{2} \) |
| 29 | \( 1 + 3.32iT - 29T^{2} \) |
| 31 | \( 1 - 0.133iT - 31T^{2} \) |
| 37 | \( 1 + 7.98T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 - 8.50iT - 53T^{2} \) |
| 59 | \( 1 + 3.09T + 59T^{2} \) |
| 61 | \( 1 + 3.22iT - 61T^{2} \) |
| 67 | \( 1 - 7.59T + 67T^{2} \) |
| 71 | \( 1 - 12.8iT - 71T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 7.08T + 83T^{2} \) |
| 89 | \( 1 + 4.95T + 89T^{2} \) |
| 97 | \( 1 + 4.99iT - 97T^{2} \) |
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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
−9.634759977621131234512595760759, −9.170140657379277151804658855050, −7.63691458649320197544545469585, −7.11014224362119177277801261167, −6.09913477529025938524823793095, −5.32895247043829658286548825793, −4.19053403468781957040128320918, −2.98157680649391592657200174372, −2.18252963359565857282958563254, −1.21673179123893456015226516535,
1.40160227074533040789698173336, 2.55655671357372906253134102982, 3.62995129519669060413959403836, 5.18660214128401379260052744723, 5.88838059205816204087056816047, 6.19528119016513792367028604071, 7.30567063070022031582266048166, 8.108422496916941645291333792794, 8.755562945230656692391914733219, 9.813884399225296598973956829239
