| L(s) = 1 | − 8·3-s − 16·4-s + 37·9-s + 128·12-s + 192·16-s + 74·25-s − 80·27-s − 680·31-s − 592·36-s + 868·37-s − 1.53e3·48-s + 686·49-s − 2.04e3·64-s + 832·67-s − 592·75-s − 359·81-s + 5.44e3·93-s − 68·97-s − 1.18e3·100-s + 2.34e3·103-s + 1.28e3·108-s − 6.94e3·111-s − 1.33e3·121-s + 1.08e4·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 2·4-s + 1.37·9-s + 3.07·12-s + 3·16-s + 0.591·25-s − 0.570·27-s − 3.93·31-s − 2.74·36-s + 3.85·37-s − 4.61·48-s + 2·49-s − 4·64-s + 1.51·67-s − 0.911·75-s − 0.492·81-s + 6.06·93-s − 0.0711·97-s − 1.18·100-s + 2.24·103-s + 1.14·108-s − 5.93·111-s − 121-s + 7.87·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4097990180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4097990180\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )( 1 + 108 T + p^{3} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 340 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 434 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 738 T + p^{3} T^{2} )( 1 + 738 T + p^{3} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )( 1 + 720 T + p^{3} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 612 T + p^{3} T^{2} )( 1 + 612 T + p^{3} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1674 T + p^{3} T^{2} )( 1 + 1674 T + p^{3} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
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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
−16.84554703144058556385728820735, −16.37553697149197884722126398811, −15.26261514956938715094399287408, −14.53693040595190469056042496956, −14.27087048812727286956085780821, −13.06046196518666073324503886292, −13.01567201937946566267444944278, −12.48522507971855363441782072949, −11.54873727461745405209685470654, −10.90826767119596348071720584485, −10.29921769304963799277093712860, −9.370413656075901871775726114474, −9.142613667290219725271417935311, −8.035652499512990361122647576289, −7.22695323575738714199770440192, −5.88253718205952308588782110229, −5.49093125966239897744959337270, −4.61241614486548268154450395183, −3.84731316563188923300606049744, −0.68018919522407504636672132604,
0.68018919522407504636672132604, 3.84731316563188923300606049744, 4.61241614486548268154450395183, 5.49093125966239897744959337270, 5.88253718205952308588782110229, 7.22695323575738714199770440192, 8.035652499512990361122647576289, 9.142613667290219725271417935311, 9.370413656075901871775726114474, 10.29921769304963799277093712860, 10.90826767119596348071720584485, 11.54873727461745405209685470654, 12.48522507971855363441782072949, 13.01567201937946566267444944278, 13.06046196518666073324503886292, 14.27087048812727286956085780821, 14.53693040595190469056042496956, 15.26261514956938715094399287408, 16.37553697149197884722126398811, 16.84554703144058556385728820735
