| L(s) = 1 | + 4.81i·3-s + 8.33·5-s − 2.64i·7-s − 14.1·9-s − 17.2i·11-s − 17.1·13-s + 40.1i·15-s − 24.3·17-s − 4.81i·19-s + 12.7·21-s + 18.9i·23-s + 44.4·25-s − 24.8i·27-s − 34.2·29-s − 59.0i·31-s + ⋯ |
| L(s) = 1 | + 1.60i·3-s + 1.66·5-s − 0.377i·7-s − 1.57·9-s − 1.56i·11-s − 1.31·13-s + 2.67i·15-s − 1.43·17-s − 0.253i·19-s + 0.606·21-s + 0.824i·23-s + 1.77·25-s − 0.920i·27-s − 1.18·29-s − 1.90i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6459868748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6459868748\) |
| \(L(2)\) |
|
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 | \( 1 - 4.81iT - 9T^{2} \) |
| 5 | \( 1 - 8.33T + 25T^{2} \) |
| 11 | \( 1 + 17.2iT - 121T^{2} \) |
| 13 | \( 1 + 17.1T + 169T^{2} \) |
| 17 | \( 1 + 24.3T + 289T^{2} \) |
| 19 | \( 1 + 4.81iT - 361T^{2} \) |
| 23 | \( 1 - 18.9iT - 529T^{2} \) |
| 29 | \( 1 + 34.2T + 841T^{2} \) |
| 31 | \( 1 + 59.0iT - 961T^{2} \) |
| 37 | \( 1 + 50.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 6.02iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 21.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 1.83T + 2.80e3T^{2} \) |
| 59 | \( 1 + 58.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 7.41T + 3.72e3T^{2} \) |
| 67 | \( 1 - 30.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 25.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 68.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 16.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 82.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 66.6T + 9.40e3T^{2} \) |
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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
−9.188115004019690708881741114828, −8.544736366956501350344089456168, −7.23252521787278104610447851998, −6.15376328269923836448595607191, −5.54034304893492922702208774425, −4.89705336833899490430052308473, −3.93658114048052293383090833862, −2.95457206906702567571615371892, −2.00023257974835355282757459670, −0.14030865193601116760410868968,
1.67142551952552828144909721141, 2.01130832058877585865176793529, 2.72130960370990613712320769185, 4.70796569316440078143285361951, 5.37368155865710090415186021898, 6.34154223712869733414555358049, 6.93972124740246037314739901578, 7.33724268231422183682281583026, 8.597351276822672824272341256664, 9.177371325785314239873399589293
