| L(s) = 1 | + (−0.802 − 1.00i)2-s + (−1.70 + 0.822i)3-s + (1.41 − 6.18i)4-s + (−15.9 + 7.67i)5-s + (2.19 + 1.05i)6-s + (0.116 − 18.5i)7-s + (−16.6 + 8.01i)8-s + (−14.5 + 18.2i)9-s + (20.5 + 9.88i)10-s + (−33.8 − 42.5i)11-s + (2.67 + 11.7i)12-s + (13.9 + 17.4i)13-s + (−18.7 + 14.7i)14-s + (20.9 − 26.2i)15-s + (−24.2 − 11.6i)16-s + (−4.20 − 18.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.283 − 0.355i)2-s + (−0.328 + 0.158i)3-s + (0.176 − 0.772i)4-s + (−1.42 + 0.686i)5-s + (0.149 + 0.0720i)6-s + (0.00631 − 0.999i)7-s + (−0.735 + 0.354i)8-s + (−0.540 + 0.677i)9-s + (0.648 + 0.312i)10-s + (−0.929 − 1.16i)11-s + (0.0643 + 0.282i)12-s + (0.297 + 0.372i)13-s + (−0.357 + 0.281i)14-s + (0.360 − 0.451i)15-s + (−0.379 − 0.182i)16-s + (−0.0599 − 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.115i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0186676 - 0.322626i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0186676 - 0.322626i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.116 + 18.5i)T \) |
| good | 2 | \( 1 + (0.802 + 1.00i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (1.70 - 0.822i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (15.9 - 7.67i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (33.8 + 42.5i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-13.9 - 17.4i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (4.20 + 18.4i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 - 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-15.3 + 67.1i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-4.70 - 20.6i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (5.03 + 22.0i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (270. - 130. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (354. + 170. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-69.1 - 86.6i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-144. + 634. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (-142. - 68.7i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-14.7 - 64.6i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 - 755.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (93.2 - 408. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (57.6 - 72.2i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + 647.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-285. + 358. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-329. + 413. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−14.56273337791191315539520872644, −13.62980017379798969132368740276, −11.48648210242345620095500013255, −11.15616359856805014288741523498, −10.22692585941689034947127992841, −8.311979377332675089816080359265, −7.06530690607846944877629623855, −5.28355074572039795275807765344, −3.29126334829009694357633871791, −0.27139308700326836797288742508,
3.38660880989070452280915997390, 5.32976138052056394886347595177, 7.25279229984607004663946123688, 8.177885101112110799795671264055, 9.256052922229210954031750281397, 11.50117044587126795612827555089, 12.13964356436819693369194961034, 12.85964154021383181070057801243, 15.21206098314804812213261713863, 15.59658576563444405075546497134
