L(s) = 1 | − 103.·5-s − 619.·7-s + 57.6·11-s − 481. i·13-s + 2.33e3·17-s + (4.35e3 + 5.29e3i)19-s + 8.69e3·23-s − 4.92e3·25-s − 4.60e4i·29-s − 3.67e4i·31-s + 6.40e4·35-s − 4.04e4i·37-s − 6.15e4i·41-s − 6.43e4·43-s + 9.39e4·47-s + ⋯ |
L(s) = 1 | − 0.827·5-s − 1.80·7-s + 0.0432·11-s − 0.219i·13-s + 0.476·17-s + (0.635 + 0.772i)19-s + 0.714·23-s − 0.315·25-s − 1.88i·29-s − 1.23i·31-s + 1.49·35-s − 0.798i·37-s − 0.892i·41-s − 0.809·43-s + 0.905·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1168000457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1168000457\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-4.35e3 - 5.29e3i)T \) |
good | 5 | \( 1 + 103.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 619.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 57.6T + 1.77e6T^{2} \) |
| 13 | \( 1 + 481. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 2.33e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 8.69e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 4.60e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.67e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.04e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 6.15e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.43e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 9.39e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 9.78e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.32e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.92e3T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.78e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.51e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.69e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.47e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 4.95e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.35e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 7.59e4iT - 8.32e11T^{2} \) |
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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
−9.791981112730930892586788708426, −9.169819876091509385407320337488, −7.955580098371709386358709021120, −7.32342066380330798915458710624, −6.28965681712601665845493827109, −5.57408931895329874739008556814, −4.03380138243028353419337080765, −3.50191829010418834532498337208, −2.45861985805907694495967273491, −0.71373002573891803641291108645,
0.03549545877075739874862473386, 1.18621931000747620099253272479, 3.03957143162234349480338431496, 3.33112158016913325017355726633, 4.59039745697829376699109497428, 5.70107720198165536167658794702, 6.85028909467356050975727970735, 7.18199150969952864920705057690, 8.487337433193803164389398989330, 9.256911121583586675567438660649