| L(s) = 1 | − 126·3-s − 128·4-s + 5.57e3·7-s + 9.31e3·9-s + 1.61e4·12-s − 2.63e4·13-s + 1.63e4·16-s + 2.88e5·19-s − 7.02e5·21-s + 4.48e5·25-s − 3.47e5·27-s − 7.13e5·28-s + 1.45e6·31-s − 1.19e6·36-s − 3.92e6·37-s + 3.31e6·39-s − 1.56e5·43-s − 2.06e6·48-s + 1.17e7·49-s + 3.36e6·52-s − 3.62e7·57-s + 3.51e7·61-s + 5.19e7·63-s − 2.09e6·64-s − 3.42e7·67-s + 5.62e7·73-s − 5.64e7·75-s + ⋯ |
| L(s) = 1 | − 1.55·3-s − 1/2·4-s + 2.32·7-s + 1.41·9-s + 7/9·12-s − 0.920·13-s + 1/4·16-s + 2.20·19-s − 3.60·21-s + 1.14·25-s − 0.652·27-s − 1.16·28-s + 1.57·31-s − 0.709·36-s − 2.09·37-s + 1.43·39-s − 0.0457·43-s − 0.388·48-s + 2.03·49-s + 0.460·52-s − 3.43·57-s + 2.53·61-s + 3.29·63-s − 1/8·64-s − 1.70·67-s + 1.98·73-s − 1.78·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.054321099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054321099\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| 3 | $C_2$ | \( 1 + 14 p^{2} T + p^{8} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 448322 T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 398 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 74615230 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 13150 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9544036610 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 144002 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 154186508930 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 606859926722 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 728738 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1964446 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14997407035010 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 78142 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 35234322443522 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 124246846237250 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 268618401162050 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 17578274 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 17136766 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 621182343784322 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 28139330 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 9182498 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3083295701563966 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1271157775602050 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 128722558 T + p^{8} T^{2} )^{2} \) |
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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
−22.29562385070242310283253082095, −20.89584655302333174624205760107, −20.71922738888534525984863229335, −19.27614328231768897055248638516, −18.18511240999852865890420946834, −17.83556206177773251820734599158, −17.38711976545548378528274612167, −16.58171257300205966983593566690, −15.50932813861659101071182746475, −14.45977038491012259071190772770, −13.83286907179083159349749942332, −12.26671734216539629219257399060, −11.73615780025091426231820046210, −10.99764242447386960739802118707, −9.952645771016127759046600591714, −8.338277152980955213364917201137, −7.17854412970097166907759560186, −5.20770109877216866850706186223, −4.89825008444268207342800220819, −1.11307400118176072275291948040,
1.11307400118176072275291948040, 4.89825008444268207342800220819, 5.20770109877216866850706186223, 7.17854412970097166907759560186, 8.338277152980955213364917201137, 9.952645771016127759046600591714, 10.99764242447386960739802118707, 11.73615780025091426231820046210, 12.26671734216539629219257399060, 13.83286907179083159349749942332, 14.45977038491012259071190772770, 15.50932813861659101071182746475, 16.58171257300205966983593566690, 17.38711976545548378528274612167, 17.83556206177773251820734599158, 18.18511240999852865890420946834, 19.27614328231768897055248638516, 20.71922738888534525984863229335, 20.89584655302333174624205760107, 22.29562385070242310283253082095
