| L(s) = 1 | − 1.55e3·3-s − 1.63e4·4-s + 8.36e5·9-s + 2.55e7·12-s + 2.01e8·16-s − 2.35e9·25-s + 1.18e9·27-s − 4.49e9·31-s − 1.36e10·36-s − 6.20e10·37-s − 3.13e11·48-s + 1.93e11·49-s − 2.19e12·64-s + 1.76e12·67-s + 3.66e12·75-s − 3.17e12·81-s + 7.01e12·93-s + 2.70e13·97-s + 3.85e13·100-s − 2.58e13·103-s − 1.93e13·108-s + 9.66e13·111-s − 3.45e13·121-s + 7.37e13·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.23·3-s − 2·4-s + 0.524·9-s + 2.46·12-s + 3·16-s − 1.92·25-s + 0.587·27-s − 0.910·31-s − 1.04·36-s − 3.97·37-s − 3.70·48-s + 2·49-s − 4·64-s + 2.38·67-s + 2.38·75-s − 1.24·81-s + 1.12·93-s + 3.29·97-s + 3.85·100-s − 2.13·103-s − 1.17·108-s + 4.90·111-s − 121-s + 1.82·124-s + 1.57·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.09602167967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09602167967\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + 1559 T + p^{13} T^{2} \) |
| 11 | $C_2$ | \( 1 + p^{13} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 9357 T + p^{13} T^{2} )( 1 + 9357 T + p^{13} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 173436111 T + p^{13} T^{2} )( 1 + 173436111 T + p^{13} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2249175235 T + p^{13} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 31003015687 T + p^{13} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 141983983452 T + p^{13} T^{2} )( 1 + 141983983452 T + p^{13} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 222700561194 T + p^{13} T^{2} )( 1 + 222700561194 T + p^{13} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 616566548505 T + p^{13} T^{2} )( 1 + 616566548505 T + p^{13} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 882597984827 T + p^{13} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 1570954867563 T + p^{13} T^{2} )( 1 + 1570954867563 T + p^{13} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1687040166249 T + p^{13} T^{2} )( 1 + 1687040166249 T + p^{13} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13535202283217 T + p^{13} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.27980786822009530124951097481, −13.44760463433760786542573947250, −12.87948134217231944858764466347, −12.03546762809334220186518885244, −12.03073868473521818176276906138, −10.82459645930067333039144275020, −10.33311495922246370968792753785, −9.759205576772203577894806135620, −8.982208996465235188763532929026, −8.542895907893248251540293739343, −7.70724359493300645412553303001, −6.82749060102880452186584621825, −5.73366384361232443659775020728, −5.42189027181009026483887115702, −4.86241137586510574686327999233, −3.91587692554119650993763046213, −3.53231735647110935622545741244, −1.89483211992879642234416268294, −0.912771842072197955834335621671, −0.13754498571211129285256087276,
0.13754498571211129285256087276, 0.912771842072197955834335621671, 1.89483211992879642234416268294, 3.53231735647110935622545741244, 3.91587692554119650993763046213, 4.86241137586510574686327999233, 5.42189027181009026483887115702, 5.73366384361232443659775020728, 6.82749060102880452186584621825, 7.70724359493300645412553303001, 8.542895907893248251540293739343, 8.982208996465235188763532929026, 9.759205576772203577894806135620, 10.33311495922246370968792753785, 10.82459645930067333039144275020, 12.03073868473521818176276906138, 12.03546762809334220186518885244, 12.87948134217231944858764466347, 13.44760463433760786542573947250, 14.27980786822009530124951097481
