| L(s) = 1 | − 15.5·2-s + 179.·4-s − 25.8i·5-s − 1.79e3·8-s + 402. i·10-s − 623.·11-s + 3.25e3i·13-s + 1.65e4·16-s + 317. i·17-s − 6.00e3i·19-s − 4.62e3i·20-s + 9.73e3·22-s + 42.0·23-s + 1.49e4·25-s − 5.07e4i·26-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.80·4-s − 0.206i·5-s − 3.50·8-s + 0.402i·10-s − 0.468·11-s + 1.48i·13-s + 4.04·16-s + 0.0646i·17-s − 0.874i·19-s − 0.578i·20-s + 0.913·22-s + 0.00345·23-s + 0.957·25-s − 2.89i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3336547887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3336547887\) |
| \(L(4)\) |
|
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 | \( 1 + 15.5T + 64T^{2} \) |
| 5 | \( 1 + 25.8iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 623.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 3.25e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 317. iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 6.00e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 42.0T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.42e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.02e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.73e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.00e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.78e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 6.58e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 9.92e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.00e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.32e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.10e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.00e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 5.53e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 1.04e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 7.12e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.90e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.07e6iT - 8.32e11T^{2} \) |
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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
−10.28165240277142353533993325478, −9.427814036149295743777483631672, −8.772373830108724084591936709608, −8.039902167290185136878356627660, −6.92504240698899475366691246790, −6.46764604115436180493173970130, −4.89121246457336784619899823548, −3.07172291520543726443467431532, −2.00096200222359678220621917822, −1.00131108361501314413192552644,
0.17345165329660512915934677086, 1.11563024421431885090606719856, 2.39398610600106983733378741180, 3.28699091499147807154134122841, 5.43057420125611907023031019916, 6.40940005839775456766602942543, 7.41087497696289980668365057630, 8.096862668296754154576847453564, 8.777530449023176118626910594863, 9.926401499336074858184172689105
