L(s) = 1 | + (−2.12 + 0.707i)5-s − 5.65i·11-s + (3 + 3i)13-s − 4i·19-s + (2.82 − 2.82i)23-s + (3.99 − 3i)25-s − 1.41·29-s + 8·31-s + (7 − 7i)37-s − 1.41i·41-s + (−4 − 4i)43-s + (2.82 + 2.82i)47-s + 7i·49-s + (8.48 − 8.48i)53-s + (4.00 + 12i)55-s + ⋯ |
L(s) = 1 | + (−0.948 + 0.316i)5-s − 1.70i·11-s + (0.832 + 0.832i)13-s − 0.917i·19-s + (0.589 − 0.589i)23-s + (0.799 − 0.600i)25-s − 0.262·29-s + 1.43·31-s + (1.15 − 1.15i)37-s − 0.220i·41-s + (−0.609 − 0.609i)43-s + (0.412 + 0.412i)47-s + i·49-s + (1.16 − 1.16i)53-s + (0.539 + 1.61i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10736 - 0.499378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10736 - 0.499378i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.12 - 0.707i)T \) |
good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-7 + 7i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (4 + 4i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.82 - 2.82i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + (-8 + 8i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 + (11.3 - 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 5i)T - 97iT^{2} \) |
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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
−10.63978753761169458267835070963, −9.177789022947758571458135912686, −8.602895315106921956134250977960, −7.78775688415809139869635752466, −6.71611463258617931191290947488, −6.01964532277683751984144589783, −4.66186777228782988772276985448, −3.70618725591907519478432028569, −2.76833326766280947840355657379, −0.73658646167168592715825000962,
1.32257312973582483185642431016, 3.00958359936662880154101299187, 4.14197461347787504084738070962, 4.90978161545585644178386950186, 6.10976704352584985003153452391, 7.24377493171804238666547475161, 7.88742062782318431241834410361, 8.680288406049595226493407106559, 9.776125886062915908780047517531, 10.43318199797163184854842212190