| L(s) = 1 | − 1.05i·2-s + 0.889·4-s + (−1.25 + 1.85i)5-s − 0.431i·7-s − 3.04i·8-s + (1.95 + 1.32i)10-s − 2.30·11-s + i·13-s − 0.454·14-s − 1.43·16-s − 4.29i·17-s + 1.06·19-s + (−1.11 + 1.64i)20-s + 2.43i·22-s − 4.55i·23-s + ⋯ |
| L(s) = 1 | − 0.745i·2-s + 0.444·4-s + (−0.560 + 0.828i)5-s − 0.162i·7-s − 1.07i·8-s + (0.617 + 0.417i)10-s − 0.695·11-s + 0.277i·13-s − 0.121·14-s − 0.357·16-s − 1.04i·17-s + 0.243·19-s + (−0.249 + 0.368i)20-s + 0.518i·22-s − 0.949i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.393608518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393608518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.25 - 1.85i)T \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 + 1.05iT - 2T^{2} \) |
| 7 | \( 1 + 0.431iT - 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 17 | \( 1 + 4.29iT - 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 + 4.55iT - 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 2.73iT - 37T^{2} \) |
| 41 | \( 1 - 2.78T + 41T^{2} \) |
| 43 | \( 1 + 5.43iT - 43T^{2} \) |
| 47 | \( 1 + 2.24iT - 47T^{2} \) |
| 53 | \( 1 + 3.48iT - 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 + 2.19iT - 67T^{2} \) |
| 71 | \( 1 + 0.423T + 71T^{2} \) |
| 73 | \( 1 + 9.90iT - 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 - 1.05iT - 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 - 1.34iT - 97T^{2} \) |
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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
−9.260100205258586675972805182696, −8.126268350036554403485564768284, −7.32110885415365488112102135789, −6.85186967692824544909646589729, −5.90541597593595481957625349436, −4.68171168677207373686670713674, −3.72030541517423117023785759317, −2.87066200105491675465397884892, −2.17478976327707152575139882117, −0.51647588762013896015319718893,
1.38545144571489673628732647672, 2.66646505290790041707389653787, 3.81847664190818498833892403208, 4.91526901134279741962169102246, 5.59342971973232731324658587958, 6.30712487607424590230351425108, 7.43376831372237433561671773206, 7.87845253676215893758602887932, 8.505275092354838936820856738399, 9.355425220153223681955176630593
