| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4838597498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4838597498\) |
| \(L(2)\) |
|
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 | \( 1 + 52.0T + 9.40e3T^{2} \) |
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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
−9.351523658829934716769092325599, −8.353319371671937867974558184703, −7.62722890299110625280431057647, −6.83914342040156838464146359961, −5.60939822222885463442550463584, −4.54314221208725155735127429604, −4.32345159836774838574443546391, −3.47113463113718698178675926820, −2.33459096578820220614016507060, −0.20461327754947243875761574581,
0.63418125394111519027939176801, 2.17085790568662205139189884064, 2.72591740598226300167890410531, 4.18749273213396704405910489857, 4.96650437050926351731129412802, 6.20583097664675079093061156631, 6.95098583465109543004077901748, 7.36964282255535902272884941435, 8.207689127432983222262043238687, 8.753128355288899694339391157048
