L(s) = 1 | − 5-s + (2.56 − 0.648i)7-s + 1.29i·11-s + 3.13i·13-s + 5.53·17-s − 7.37i·19-s − 1.83i·23-s + 25-s + 1.83i·29-s − 10.4i·31-s + (−2.56 + 0.648i)35-s − 10.6·37-s + 3.13·41-s − 3.53·43-s + 10.7·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + (0.969 − 0.245i)7-s + 0.390i·11-s + 0.868i·13-s + 1.34·17-s − 1.69i·19-s − 0.382i·23-s + 0.200·25-s + 0.340i·29-s − 1.87i·31-s + (−0.433 + 0.109i)35-s − 1.75·37-s + 0.488·41-s − 0.539·43-s + 1.57·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.992452295\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.992452295\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + (-2.56 + 0.648i)T \) |
good | 11 | \( 1 - 1.29iT - 11T^{2} \) |
| 13 | \( 1 - 3.13iT - 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 7.37iT - 19T^{2} \) |
| 23 | \( 1 + 1.83iT - 23T^{2} \) |
| 29 | \( 1 - 1.83iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.42iT - 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 + 4.88iT - 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 - 7.37iT - 71T^{2} \) |
| 73 | \( 1 - 3.40iT - 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 8.09iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.132706589856015644813916668628, −7.30016312608149572597995930946, −7.05373872307757437272304321010, −5.92644439206988639048057038665, −5.08846613469399406175313882492, −4.47342764019601397554077994334, −3.79612862023185805100002329326, −2.69115772880136597406640124941, −1.76825182063481784367282537826, −0.64784742988556566116388412625,
1.00539830972395929301517112195, 1.86104305615810810756772896453, 3.20738146973261972017248888159, 3.62683117026749716498603404345, 4.73740991422391272834007135073, 5.49438509202727807900368211644, 5.87681864974266584749710088447, 7.10261743310150880099371654643, 7.70714393039866383851978988934, 8.292218345021052429858515703283