L(s) = 1 | − 0.332i·2-s + 1.45·3-s + 1.88·4-s − 1.44i·5-s − 0.485i·6-s − 1.29i·8-s − 0.868·9-s − 0.480·10-s − 5.95i·11-s + 2.75·12-s + (1.88 + 3.07i)13-s − 2.11i·15-s + 3.34·16-s − 4.32·17-s + 0.288i·18-s − 1.95i·19-s + ⋯ |
L(s) = 1 | − 0.235i·2-s + 0.842·3-s + 0.944·4-s − 0.646i·5-s − 0.198i·6-s − 0.457i·8-s − 0.289·9-s − 0.151·10-s − 1.79i·11-s + 0.796·12-s + (0.524 + 0.851i)13-s − 0.544i·15-s + 0.837·16-s − 1.04·17-s + 0.0680i·18-s − 0.449i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02795 - 1.13326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02795 - 1.13326i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-1.88 - 3.07i)T \) |
good | 2 | \( 1 + 0.332iT - 2T^{2} \) |
| 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 + 1.44iT - 5T^{2} \) |
| 11 | \( 1 + 5.95iT - 11T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 + 1.95iT - 19T^{2} \) |
| 23 | \( 1 - 0.540T + 23T^{2} \) |
| 29 | \( 1 - 7.15T + 29T^{2} \) |
| 31 | \( 1 - 6.10iT - 31T^{2} \) |
| 37 | \( 1 - 8.02iT - 37T^{2} \) |
| 41 | \( 1 - 7.55iT - 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 - 6.26iT - 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 + 0.851iT - 59T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 - 0.987iT - 67T^{2} \) |
| 71 | \( 1 + 3.76iT - 71T^{2} \) |
| 73 | \( 1 + 9.13iT - 73T^{2} \) |
| 79 | \( 1 + 0.131T + 79T^{2} \) |
| 83 | \( 1 + 2.66iT - 83T^{2} \) |
| 89 | \( 1 - 9.71iT - 89T^{2} \) |
| 97 | \( 1 + 6.58iT - 97T^{2} \) |
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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
−10.65125317819047033062139166464, −9.350681461927720047273174873219, −8.562183127373869299809922630308, −8.219508976203266144415900420230, −6.74150469174236634979403305615, −6.15258449824332125385342161073, −4.79301955599348282989734743743, −3.40246319201925240981682139158, −2.71242691506721898770817309831, −1.25797718227731155594466908914,
2.06680643627822962523039156360, 2.74027316922378764702108304523, 3.93517832464272092468138988249, 5.37954049021917531869701435984, 6.51311017772241456017931085094, 7.19773275294271035104493364989, 7.973933103355446169060684036081, 8.835489241041526957626304482274, 9.973709435541264428543340877161, 10.62754673927414002124579732452