| L(s) = 1 | + (−16.9 + 29.4i)2-s + (797. − 1.38e3i)3-s + (3.51e3 + 6.09e3i)4-s − 2.82e4·5-s + (2.71e4 + 4.69e4i)6-s + (1.77e5 + 3.06e5i)7-s − 5.17e5·8-s + (−4.75e5 − 8.24e5i)9-s + (4.79e5 − 8.29e5i)10-s + (2.48e6 − 4.30e6i)11-s + 1.12e7·12-s + (1.11e7 + 1.33e7i)13-s − 1.20e7·14-s + (−2.25e7 + 3.89e7i)15-s + (−2.00e7 + 3.47e7i)16-s + (5.35e7 + 9.27e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.187 + 0.325i)2-s + (0.631 − 1.09i)3-s + (0.429 + 0.743i)4-s − 0.807·5-s + (0.237 + 0.410i)6-s + (0.569 + 0.985i)7-s − 0.697·8-s + (−0.298 − 0.517i)9-s + (0.151 − 0.262i)10-s + (0.422 − 0.732i)11-s + 1.08·12-s + (0.639 + 0.769i)13-s − 0.427·14-s + (−0.510 + 0.883i)15-s + (−0.298 + 0.517i)16-s + (0.538 + 0.932i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.81115 + 0.930103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81115 + 0.930103i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.11e7 - 1.33e7i)T \) |
| good | 2 | \( 1 + (16.9 - 29.4i)T + (-4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-797. + 1.38e3i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + 2.82e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + (-1.77e5 - 3.06e5i)T + (-4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (-2.48e6 + 4.30e6i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (-5.35e7 - 9.27e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-1.61e8 - 2.80e8i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-2.62e8 + 4.55e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + (-6.01e8 + 1.04e9i)T + (-5.13e18 - 8.88e18i)T^{2} \) |
| 31 | \( 1 - 3.17e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + (9.24e9 - 1.60e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (1.08e10 - 1.88e10i)T + (-4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (2.57e10 + 4.46e10i)T + (-8.59e20 + 1.48e21i)T^{2} \) |
| 47 | \( 1 - 1.84e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.97e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + (7.27e10 + 1.26e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (3.18e11 + 5.50e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-3.30e11 + 5.73e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (-8.77e11 - 1.52e12i)T + (-5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 + 1.33e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.34e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.49e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (3.25e12 - 5.63e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (3.39e11 + 5.88e11i)T + (-3.36e25 + 5.82e25i)T^{2} \) |
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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
−16.79315572564353976177233768630, −15.48990840211563950233176004973, −14.06391814567653354628904693421, −12.37899716881819557947264515762, −11.61756008261457035652367981912, −8.490266658508391966653854447147, −7.987145635478967199451124067291, −6.38310551255692624887666179316, −3.40654012278861641354529017888, −1.68903865246610400176757845930,
0.919189074850343573993118093333, 3.27062996251765495021699544165, 4.78228815372283619526607599817, 7.37188553443749618957048234909, 9.282635526467940629300982350981, 10.47059876727197677300485747304, 11.59653971205263521336428688192, 14.02247573398477407263445008174, 15.22236506188346913694020864512, 15.91884626781929775694736091197
