| L(s) = 1 | + 5.65·2-s + 31.8i·3-s + 32.0·4-s − 129. i·5-s + 180. i·6-s + 181.·8-s − 287.·9-s − 730. i·10-s + 2.37e3·11-s + 1.02e3i·12-s + 820. i·13-s + 4.11e3·15-s + 1.02e3·16-s − 2.63e3i·17-s − 1.62e3·18-s + 5.55e3i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.18i·3-s + 0.500·4-s − 1.03i·5-s + 0.834i·6-s + 0.353·8-s − 0.394·9-s − 0.730i·10-s + 1.78·11-s + 0.590i·12-s + 0.373i·13-s + 1.21·15-s + 0.250·16-s − 0.536i·17-s − 0.278·18-s + 0.810i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.18094 + 1.18593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.18094 + 1.18593i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 31.8iT - 729T^{2} \) |
| 5 | \( 1 + 129. iT - 1.56e4T^{2} \) |
| 11 | \( 1 - 2.37e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 820. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.63e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.55e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.13e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 3.28e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.69e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 2.17e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 8.58e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.13e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 3.85e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 3.32e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.34e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.68e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.22e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.25e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.07e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.39e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 2.24e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 3.83e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.26e4iT - 8.32e11T^{2} \) |
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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
−12.76254994728925279536858346856, −11.90046487659916191762047672773, −10.83351356419199107640277370868, −9.456317757361505549570622031089, −8.874218727059248834692764100682, −6.99040577939503031742706854868, −5.47590609639659484430349269416, −4.44716147124685369100637714148, −3.62542378903969993238997353671, −1.35727151488341893449551637675,
1.19271860961090183350302941125, 2.60613574951610796281916221591, 4.04100611062674829161185961032, 6.05634794964856201339693192248, 6.79234516241518337092175636030, 7.63283653899079796439415605824, 9.325553616501600985696200279340, 10.94590954882672162668522690766, 11.68360935822968870678063013730, 12.78124877968528462223052962702
