| L(s) = 1 | − 0.585i·3-s − 9.03·5-s − 2.64i·7-s + 8.65·9-s + 12.4i·11-s − 9.03·13-s + 5.29i·15-s + 12.3·17-s − 28.8i·19-s − 1.54·21-s + 24.6i·23-s + 56.5·25-s − 10.3i·27-s + 22.4·29-s − 16.7i·31-s + ⋯ |
| L(s) = 1 | − 0.195i·3-s − 1.80·5-s − 0.377i·7-s + 0.961·9-s + 1.13i·11-s − 0.694·13-s + 0.352i·15-s + 0.726·17-s − 1.51i·19-s − 0.0738·21-s + 1.07i·23-s + 2.26·25-s − 0.383i·27-s + 0.774·29-s − 0.541i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1306118802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1306118802\) |
| \(L(2)\) |
|
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 | \( 1 + 0.585iT - 9T^{2} \) |
| 5 | \( 1 + 9.03T + 25T^{2} \) |
| 11 | \( 1 - 12.4iT - 121T^{2} \) |
| 13 | \( 1 + 9.03T + 169T^{2} \) |
| 17 | \( 1 - 12.3T + 289T^{2} \) |
| 19 | \( 1 + 28.8iT - 361T^{2} \) |
| 23 | \( 1 - 24.6iT - 529T^{2} \) |
| 29 | \( 1 - 22.4T + 841T^{2} \) |
| 31 | \( 1 + 16.7iT - 961T^{2} \) |
| 37 | \( 1 + 16.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 6.97T + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 6.19iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.01T + 2.80e3T^{2} \) |
| 59 | \( 1 - 30.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 78.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 17.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 46.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 81.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 40.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 164.T + 9.40e3T^{2} \) |
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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
−8.550786682878634080436734640702, −7.54380505727756542997504464114, −7.36148955048864749376466054296, −6.76182600647346011316667636885, −5.11257374911834152371835672057, −4.45342567598203053133029703389, −3.82819319209372386272218765406, −2.72058826377050869549971784177, −1.24207954382935543286620492625, −0.04148731970417866683723149510,
1.20820512884525231893507062711, 2.93038123619989342506128500219, 3.68482930758597084771049067155, 4.40845857312911585360351787929, 5.28054204357740376921848418492, 6.43426277562432715606004590143, 7.24650579673244309554904520685, 8.147231607066325946240335967848, 8.310300025852285562417381901916, 9.477812913283216876761318836487
