L(s) = 1 | + 1.73i·3-s − 1.73i·7-s − 1.99·9-s − 1.73i·11-s + 2.99·21-s + 25-s − 1.73i·27-s + 2.99·33-s + 37-s + 41-s − 1.73i·47-s − 1.99·49-s − 53-s + 3.46i·63-s + 1.73i·71-s + ⋯ |
L(s) = 1 | + 1.73i·3-s − 1.73i·7-s − 1.99·9-s − 1.73i·11-s + 2.99·21-s + 25-s − 1.73i·27-s + 2.99·33-s + 37-s + 41-s − 1.73i·47-s − 1.99·49-s − 53-s + 3.46i·63-s + 1.73i·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.087671447\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.087671447\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 1.73iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.73iT - T^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.73iT - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.73iT - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.73iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.253745414595229766746336443743, −8.558108252258844648473568049855, −7.82736739647403873484907137699, −6.76679494396556988287041100509, −5.87221609200375922492376725239, −5.01736532782025949085912310481, −4.21504248118214788918653621125, −3.63668538904622061172498963309, −2.94487597444530737984696395659, −0.78508111203247531440531445542,
1.46124396291682490439625216753, 2.30827044410402804011478442289, 2.84582421369879677083942421569, 4.58429700926321065285739997679, 5.44697350517500104266864361522, 6.26109889425609610462854014511, 6.78800415253835358858762010798, 7.70057809234086842023635370639, 8.131295299173842170384293158511, 9.163624292699498090621066515470