L(s) = 1 | + (−1.96 + 1.96i)3-s + (−0.254 + 2.22i)5-s + (−1.87 − 1.87i)7-s − 4.74i·9-s + (4.88 − 4.88i)13-s + (−3.87 − 4.87i)15-s + 2.41i·19-s + 7.36·21-s + (6.74 − 6.74i)23-s + (−4.87 − 1.12i)25-s + (3.42 + 3.42i)27-s + (4.63 − 3.68i)35-s + 19.2i·39-s + (10.5 + 1.20i)45-s + 7i·49-s + ⋯ |
L(s) = 1 | + (−1.13 + 1.13i)3-s + (−0.113 + 0.993i)5-s + (−0.707 − 0.707i)7-s − 1.58i·9-s + (1.35 − 1.35i)13-s + (−0.999 − 1.25i)15-s + 0.552i·19-s + 1.60·21-s + (1.40 − 1.40i)23-s + (−0.974 − 0.225i)25-s + (0.659 + 0.659i)27-s + (0.782 − 0.622i)35-s + 3.07i·39-s + (1.57 + 0.179i)45-s + i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9054362394\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9054362394\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.254 - 2.22i)T \) |
| 7 | \( 1 + (1.87 + 1.87i)T \) |
good | 3 | \( 1 + (1.96 - 1.96i)T - 3iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-4.88 + 4.88i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 2.41iT - 19T^{2} \) |
| 23 | \( 1 + (-6.74 + 6.74i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 + 6.47T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 8.25iT - 79T^{2} \) |
| 83 | \( 1 + (-12.2 + 12.2i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−10.28537004483072026962761949412, −9.395107387571480495005831997334, −8.279171749074187156436443119120, −7.16359945136244566953818988247, −6.28820476187032781405047901703, −5.80728484600472618083138905844, −4.65192489990083232930021520477, −3.69318461632874718489411016173, −3.06170289917278022764448738124, −0.63894087885402324468949746112,
0.945198921250729642269508848638, 1.91854399737189503265027566937, 3.59500492625356168714462192381, 4.91539054181101343907412625580, 5.63398283866769738331705109735, 6.42806103246830757796047201558, 7.00444317041011216256185486951, 8.098106057667941735027784395267, 9.028697918944278529368078067305, 9.487516992908963015819277078799