L(s) = 1 | + 1.73i·3-s − 1.99·9-s + 1.73·11-s + i·17-s + 1.73·19-s − 1.73i·27-s + 2.99i·33-s − 41-s − 49-s − 1.73·51-s + 2.99i·57-s + 1.73i·67-s − i·73-s + 0.999·81-s − 1.73i·83-s + ⋯ |
L(s) = 1 | + 1.73i·3-s − 1.99·9-s + 1.73·11-s + i·17-s + 1.73·19-s − 1.73i·27-s + 2.99i·33-s − 41-s − 49-s − 1.73·51-s + 2.99i·57-s + 1.73i·67-s − i·73-s + 0.999·81-s − 1.73i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.365094241\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.365094241\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.73iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.73iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.73iT - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + 2iT - T^{2} \) |
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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
−9.056348516839121435878073057219, −8.778749000283217275521799396292, −7.73521380678105537250780475126, −6.68403036771457460349755217084, −5.90315604120904633698518190696, −5.14298041303352628363515038931, −4.33187611646764080216131599842, −3.68559860862640257574997850971, −3.09812018185450359245456932186, −1.48833596958370965274205684930,
0.962272015099801944067239710958, 1.67225527033260316389962115265, 2.81879953781749984825649418514, 3.64970143225008491407511945110, 4.95784078782762465087232220298, 5.79172739906683840547376247573, 6.66590090308241233486679925262, 6.97505337915531153988296712410, 7.72234459521281961781775552239, 8.429389338088979554873821931944