L(s) = 1 | + 3-s + (−2 + 2i)5-s + (−1 − i)7-s + 9-s + (2 − 2i)11-s + (−2 − 3i)13-s + (−2 + 2i)15-s − 6i·17-s + (3 + 3i)19-s + (−1 − i)21-s + 8·23-s − 3i·25-s + 27-s − 2i·29-s + (1 − i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.894 + 0.894i)5-s + (−0.377 − 0.377i)7-s + 0.333·9-s + (0.603 − 0.603i)11-s + (−0.554 − 0.832i)13-s + (−0.516 + 0.516i)15-s − 1.45i·17-s + (0.688 + 0.688i)19-s + (−0.218 − 0.218i)21-s + 1.66·23-s − 0.600i·25-s + 0.192·27-s − 0.371i·29-s + (0.179 − 0.179i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.638009937\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.638009937\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (2 + 3i)T \) |
good | 5 | \( 1 + (2 - 2i)T - 5iT^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2 + 2i)T - 11iT^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + (-3 - 3i)T + 19iT^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + (-1 + i)T - 31iT^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + (-8 - 8i)T + 41iT^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + (-8 - 8i)T + 47iT^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + (-4 + 4i)T - 59iT^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + (-1 - i)T + 67iT^{2} \) |
| 71 | \( 1 + (-6 + 6i)T - 71iT^{2} \) |
| 73 | \( 1 + (9 - 9i)T - 73iT^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + (8 + 8i)T + 83iT^{2} \) |
| 89 | \( 1 + (-10 + 10i)T - 89iT^{2} \) |
| 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
−9.608195974989179540362133882656, −8.873816648654418398204549530443, −7.71739132432918913635486924143, −7.39148092861206385743316881396, −6.59019042602615605148810720955, −5.38848709065482925175767906068, −4.22535032953592961320577775724, −3.22344664850206780044766960617, −2.83383029166060534475310515946, −0.77980684202166507448610550921,
1.19321131050170795818298814961, 2.55050107259601160823664809933, 3.79851448494403413737634179329, 4.43651349613814012205082279406, 5.37423561457096111467050321642, 6.71348491411521045121601376666, 7.32441000261746697874392914319, 8.280211946432805782008733871247, 9.089405665788267403164605874498, 9.310092177068398307096915566120