L(s) = 1 | − 3.30·3-s − 13.4i·5-s − 16.0·9-s + 28.0i·11-s + 27.3i·13-s + 44.4i·15-s − 1.11i·17-s + 150.·19-s − 67.4i·23-s − 56.2·25-s + 142.·27-s + 217.·29-s + 42.9·31-s − 92.6i·33-s − 238.·37-s + ⋯ |
L(s) = 1 | − 0.635·3-s − 1.20i·5-s − 0.595·9-s + 0.768i·11-s + 0.583i·13-s + 0.765i·15-s − 0.0158i·17-s + 1.81·19-s − 0.611i·23-s − 0.449·25-s + 1.01·27-s + 1.39·29-s + 0.249·31-s − 0.488i·33-s − 1.05·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.195369297\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.195369297\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 3.30T + 27T^{2} \) |
| 5 | \( 1 + 13.4iT - 125T^{2} \) |
| 11 | \( 1 - 28.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 27.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 1.11iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 21.4iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 219. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 666.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 782.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 112. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 621. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 272. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 837. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 105.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.84e3iT - 9.12e5T^{2} \) |
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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
−9.573111851392496580881052196475, −8.902980621341704213696348619325, −8.057580726772407313603483591913, −7.00771909179380010393862548363, −6.02740550547641920901864777078, −5.02294976350909961744188135098, −4.61810925884258986371570403737, −3.09362331658298741210054665643, −1.56103476969827148048492208686, −0.44248068462268389905450696164,
0.987110009628187300307129374456, 2.87289301808168935586196800564, 3.31781422657557256347033899952, 4.98149915015054039909057360192, 5.80970335047281151669944974835, 6.52249926548057489132532854343, 7.45869052609508116384196335353, 8.308656867056799335289179052983, 9.421317273497616900296555032019, 10.37877539572093891984247452714