L(s) = 1 | − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 5-s − 2.00·6-s + 1.41i·7-s − 1.00·9-s + 1.41i·10-s − 1.41i·11-s + 1.41i·12-s + 2.00·14-s + 1.41i·15-s − 0.999·16-s + 17-s + 1.41i·18-s + 19-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 5-s − 2.00·6-s + 1.41i·7-s − 1.00·9-s + 1.41i·10-s − 1.41i·11-s + 1.41i·12-s + 2.00·14-s + 1.41i·15-s − 0.999·16-s + 17-s + 1.41i·18-s + 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6877010839\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6877010839\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 331 | \( 1 + T \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 3 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.41iT - T^{2} \) |
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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
−11.76683468859433818946477874790, −10.91083873293821841964060739790, −9.507420669366514883360334719268, −8.492519183158027149559572020791, −7.78215009456723663792192470023, −6.51880369802040179103685985516, −5.37791322093221991313937350617, −3.48887979307660269867872198718, −2.69167545616688574399335128965, −1.19567560639297745895689434972,
3.62076338683993143375358978340, 4.44967664222773039451235180042, 5.12872730530550645533283424764, 6.69627611116808674447654270290, 7.58691757887878700850819561572, 8.103718495431212188555855598732, 9.644803382779970298084804863080, 10.05421997598420128887414426285, 11.19950722450715336506096075360, 12.10255315698881699083287672092