| L(s) = 1 | + (−4.77 − 1.27i)2-s + (4.54 + 7.86i)3-s + (−34.2 − 19.7i)4-s + (−109. + 109. i)5-s + (−11.6 − 43.3i)6-s + (−504. + 135. i)7-s + (361. + 361. i)8-s + (323. − 559. i)9-s + (663. − 383. i)10-s + (246. − 919. i)11-s − 359. i·12-s + (300. + 2.17e3i)13-s + 2.58e3·14-s + (−1.36e3 − 364. i)15-s + (1.54 + 2.68i)16-s + (−5.34e3 − 3.08e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.596 − 0.159i)2-s + (0.168 + 0.291i)3-s + (−0.535 − 0.309i)4-s + (−0.877 + 0.877i)5-s + (−0.0538 − 0.200i)6-s + (−1.47 + 0.394i)7-s + (0.706 + 0.706i)8-s + (0.443 − 0.767i)9-s + (0.663 − 0.383i)10-s + (0.185 − 0.691i)11-s − 0.208i·12-s + (0.136 + 0.990i)13-s + 0.940·14-s + (−0.403 − 0.108i)15-s + (0.000378 + 0.000655i)16-s + (−1.08 − 0.628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0449915 + 0.215211i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0449915 + 0.215211i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-300. - 2.17e3i)T \) |
| good | 2 | \( 1 + (4.77 + 1.27i)T + (55.4 + 32i)T^{2} \) |
| 3 | \( 1 + (-4.54 - 7.86i)T + (-364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (109. - 109. i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + (504. - 135. i)T + (1.01e5 - 5.88e4i)T^{2} \) |
| 11 | \( 1 + (-246. + 919. i)T + (-1.53e6 - 8.85e5i)T^{2} \) |
| 17 | \( 1 + (5.34e3 + 3.08e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.06e3 - 3.95e3i)T + (-4.07e7 + 2.35e7i)T^{2} \) |
| 23 | \( 1 + (9.99e3 - 5.76e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (186. + 322. i)T + (-2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.08e4 - 1.08e4i)T - 8.87e8iT^{2} \) |
| 37 | \( 1 + (2.23e4 - 8.34e4i)T + (-2.22e9 - 1.28e9i)T^{2} \) |
| 41 | \( 1 + (-1.61e3 - 432. i)T + (4.11e9 + 2.37e9i)T^{2} \) |
| 43 | \( 1 + (8.83e4 + 5.10e4i)T + (3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (2.76e4 + 2.76e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 - 2.39e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.20e5 + 3.22e4i)T + (3.65e10 - 2.10e10i)T^{2} \) |
| 61 | \( 1 + (-3.82e4 + 6.61e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (3.54e5 + 9.50e4i)T + (7.83e10 + 4.52e10i)T^{2} \) |
| 71 | \( 1 + (1.11e5 + 4.17e5i)T + (-1.10e11 + 6.40e10i)T^{2} \) |
| 73 | \( 1 + (-3.43e5 - 3.43e5i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 + 5.59e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (5.51e4 - 5.51e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + (1.89e5 - 7.08e5i)T + (-4.30e11 - 2.48e11i)T^{2} \) |
| 97 | \( 1 + (-3.28e5 - 1.22e6i)T + (-7.21e11 + 4.16e11i)T^{2} \) |
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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
−18.99002630745060446217596343305, −18.32072516093902756586779196758, −16.32073279641383157136361579313, −15.20183171961515563360711867539, −13.66983811116506301684615556211, −11.70398011910301577808959017700, −10.03988782084036896156845757175, −8.904646485132924383134576642328, −6.67356704279308506767378270564, −3.66209977729891911120260313639,
0.20234623542748597957589959277, 4.18693168959237015533867176305, 7.29313888701180730309696394275, 8.631495046570037573677865082408, 10.15256339774363322707231863426, 12.67900780824347180402812752001, 13.21102654030892646228921867011, 15.75513084031562921180013702884, 16.56107991459155638963349349351, 17.97698837918069913144722514796
