| L(s) = 1 | − 4.56i·3-s + 5.73·5-s − 2.64i·7-s − 11.8·9-s − 1.40i·11-s + 19.0·13-s − 26.1i·15-s − 32.2·17-s − 12.5i·19-s − 12.0·21-s + 15.8i·23-s + 7.86·25-s + 13.0i·27-s + 3.29·29-s − 22.6i·31-s + ⋯ |
| L(s) = 1 | − 1.52i·3-s + 1.14·5-s − 0.377i·7-s − 1.31·9-s − 0.127i·11-s + 1.46·13-s − 1.74i·15-s − 1.89·17-s − 0.661i·19-s − 0.575·21-s + 0.690i·23-s + 0.314·25-s + 0.484i·27-s + 0.113·29-s − 0.731i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.860647547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860647547\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 + 4.56iT - 9T^{2} \) |
| 5 | \( 1 - 5.73T + 25T^{2} \) |
| 11 | \( 1 + 1.40iT - 121T^{2} \) |
| 13 | \( 1 - 19.0T + 169T^{2} \) |
| 17 | \( 1 + 32.2T + 289T^{2} \) |
| 19 | \( 1 + 12.5iT - 361T^{2} \) |
| 23 | \( 1 - 15.8iT - 529T^{2} \) |
| 29 | \( 1 - 3.29T + 841T^{2} \) |
| 31 | \( 1 + 22.6iT - 961T^{2} \) |
| 37 | \( 1 + 54.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 7.59T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 21.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 0.356T + 2.80e3T^{2} \) |
| 59 | \( 1 - 26.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 114. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 79.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 2.66T + 7.92e3T^{2} \) |
| 97 | \( 1 + 52.0T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.793819837101588036024231101256, −7.78966334203883458295057629807, −6.97620735366438902156180659742, −6.31887446222093680627953368126, −5.90759816191333823580801766251, −4.70060068522267546625199197324, −3.40273173350898997445712245009, −2.13533556105554834733756543527, −1.64469119432552745456114773721, −0.44562539874739773700159323587,
1.62245959057119261244762142305, 2.71275619799378445740869988130, 3.79303239286779236209879447220, 4.53387405404476289478787000024, 5.41281785184468768203304942885, 6.09509532394731909675727177858, 6.81599782480082977636535934574, 8.520596743872022983153169326239, 8.741251203707615091536046226997, 9.550647056281087548916024108648
