| L(s) = 1 | + 0.0974i·3-s + 3.46·5-s + 2.64i·7-s + 8.99·9-s + 2.92i·11-s − 19.1·13-s + 0.337i·15-s − 14.3·17-s + 8.09i·19-s − 0.257·21-s + 16.7i·23-s − 12.9·25-s + 1.75i·27-s − 27.1·29-s − 44.8i·31-s + ⋯ |
| L(s) = 1 | + 0.0324i·3-s + 0.693·5-s + 0.377i·7-s + 0.998·9-s + 0.266i·11-s − 1.47·13-s + 0.0225i·15-s − 0.846·17-s + 0.426i·19-s − 0.0122·21-s + 0.728i·23-s − 0.519·25-s + 0.0649i·27-s − 0.936·29-s − 1.44i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2856025437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2856025437\) |
| \(L(2)\) |
|
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 | \( 1 - 0.0974iT - 9T^{2} \) |
| 5 | \( 1 - 3.46T + 25T^{2} \) |
| 11 | \( 1 - 2.92iT - 121T^{2} \) |
| 13 | \( 1 + 19.1T + 169T^{2} \) |
| 17 | \( 1 + 14.3T + 289T^{2} \) |
| 19 | \( 1 - 8.09iT - 361T^{2} \) |
| 23 | \( 1 - 16.7iT - 529T^{2} \) |
| 29 | \( 1 + 27.1T + 841T^{2} \) |
| 31 | \( 1 + 44.8iT - 961T^{2} \) |
| 37 | \( 1 + 39.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 45.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 61.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.69T + 2.80e3T^{2} \) |
| 59 | \( 1 - 114. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 7.48T + 3.72e3T^{2} \) |
| 67 | \( 1 + 12.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 129. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 18.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 42.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 109. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 162.T + 9.40e3T^{2} \) |
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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
−9.563161533203972302183518579980, −8.945623897297846789097691142105, −7.68403822325906026831607527969, −7.24380981403544259550458120244, −6.27203846493294987376569527132, −5.42072809628424509817741917605, −4.65238300913290360085666926330, −3.70064282740834051669167952526, −2.30623700830283101811006184688, −1.74548643153941500544293985029,
0.06589966824175060890681003402, 1.57423496113320449270195715648, 2.43420834911781731783558754230, 3.66602401897134276807159273777, 4.72500232278620662236747556987, 5.25355679838230881450848401879, 6.59012002623834487440849948867, 6.92738255951249749152420985935, 7.83897728571490161184257929822, 8.804465473491534166416369254100
