L(s) = 1 | + 5.29i·3-s + 10.5i·5-s + 8·7-s − 1.00·9-s − 15.8i·11-s − 52.9i·13-s − 56.0·15-s − 14·17-s − 37.0i·19-s + 42.3i·21-s + 152·23-s + 12.9·25-s + 137. i·27-s + 158. i·29-s − 224·31-s + ⋯ |
L(s) = 1 | + 1.01i·3-s + 0.946i·5-s + 0.431·7-s − 0.0370·9-s − 0.435i·11-s − 1.12i·13-s − 0.963·15-s − 0.199·17-s − 0.447i·19-s + 0.439i·21-s + 1.37·23-s + 0.103·25-s + 0.980i·27-s + 1.01i·29-s − 1.29·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.03680 + 0.716516i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03680 + 0.716516i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 5.29iT - 27T^{2} \) |
| 5 | \( 1 - 10.5iT - 125T^{2} \) |
| 7 | \( 1 - 8T + 343T^{2} \) |
| 11 | \( 1 + 15.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 14T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 152T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 224T + 2.97e4T^{2} \) |
| 37 | \( 1 + 243. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 + 439. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 336T + 1.03e5T^{2} \) |
| 53 | \( 1 + 31.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 534. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 95.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 174. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 72T + 3.57e5T^{2} \) |
| 73 | \( 1 + 294T + 3.89e5T^{2} \) |
| 79 | \( 1 - 464T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994T + 9.12e5T^{2} \) |
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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
−16.36663995441058451432273437405, −15.22600646220450630565813851045, −14.56916251995215470821563962086, −12.97154344728769392956231004345, −11.05795936409614716996614663243, −10.48930337573917175152978976252, −8.962219872740643699774911562801, −7.17054636653215684856910850630, −5.20109583320908424907966408416, −3.32708994586277697613764332643,
1.55544138788419192924744643292, 4.72165569789740513650001362570, 6.70450696965871895365971855438, 8.064787010020897541921223108508, 9.432132866198387105181951896749, 11.42559369388467882930208929465, 12.56204586519946299862426039103, 13.38164339444501231594046082110, 14.75359118699136027758222501784, 16.31864788012553520013548286332