| L(s) = 1 | + (−32 + 55.4i)2-s + (−679. − 1.17e3i)3-s + (−2.04e3 − 3.54e3i)4-s + (−2.90e4 + 5.02e4i)5-s + 8.69e4·6-s + (−1.73e5 + 2.58e5i)7-s + 2.62e5·8-s + (−1.26e5 + 2.19e5i)9-s + (−1.85e6 − 3.21e6i)10-s + (−3.81e6 − 6.60e6i)11-s + (−2.78e6 + 4.82e6i)12-s + 1.50e7·13-s + (−8.75e6 − 1.78e7i)14-s + 7.88e7·15-s + (−8.38e6 + 1.45e7i)16-s + (1.56e7 + 2.70e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.538 − 0.932i)3-s + (−0.249 − 0.433i)4-s + (−0.830 + 1.43i)5-s + 0.761·6-s + (−0.558 + 0.829i)7-s + 0.353·8-s + (−0.0794 + 0.137i)9-s + (−0.587 − 1.01i)10-s + (−0.649 − 1.12i)11-s + (−0.269 + 0.466i)12-s + 0.864·13-s + (−0.310 − 0.635i)14-s + 1.78·15-s + (−0.125 + 0.216i)16-s + (0.157 + 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.695241 - 0.188255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.695241 - 0.188255i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (32 - 55.4i)T \) |
| 7 | \( 1 + (1.73e5 - 2.58e5i)T \) |
| good | 3 | \( 1 + (679. + 1.17e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (2.90e4 - 5.02e4i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (3.81e6 + 6.60e6i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 - 1.50e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-1.56e7 - 2.70e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-7.42e7 + 1.28e8i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-2.72e8 + 4.71e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 - 5.67e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-4.40e8 - 7.62e8i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (6.35e9 - 1.10e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 - 4.58e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 6.31e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-5.60e10 + 9.70e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-6.36e10 - 1.10e11i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (-2.26e10 - 3.91e10i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (1.50e10 - 2.61e10i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (9.77e10 + 1.69e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + 1.17e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-2.29e11 - 3.98e11i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-9.86e11 + 1.70e12i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 - 4.81e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-3.46e12 + 6.00e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + 5.74e11T + 6.73e25T^{2} \) |
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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
−16.09416958657995444880899101726, −15.16916089121805483277641026860, −13.54560354186084411371325958023, −11.88392568911464350096747361899, −10.64729222403276603462306117082, −8.388051029532871744108169701883, −6.90921027162496059674910404234, −6.04154215806733234124604822645, −3.03472302068902300112381514241, −0.48378893604349312030422742868,
0.959219610675506980943954305121, 3.89457206319466965417972139925, 4.91606615038611288384673916043, 7.80535432027321793442505395413, 9.468727572731474280686486288919, 10.61369224938419170908465791586, 12.08766286453860393362851816088, 13.24237321846308939006315231512, 15.82230007683966895925813951199, 16.32035226575674065363938380659
