| L(s) = 1 | + 4-s − 3·5-s + 5·9-s − 4·11-s + 16-s − 16·19-s − 3·20-s + 5·25-s − 12·29-s + 2·31-s + 5·36-s + 12·41-s − 4·44-s − 15·45-s + 4·49-s + 12·55-s + 18·59-s + 20·61-s + 5·64-s − 6·71-s − 16·76-s − 28·79-s − 3·80-s + 9·81-s + 6·89-s + 48·95-s − 20·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.34·5-s + 5/3·9-s − 1.20·11-s + 1/4·16-s − 3.67·19-s − 0.670·20-s + 25-s − 2.22·29-s + 0.359·31-s + 5/6·36-s + 1.87·41-s − 0.603·44-s − 2.23·45-s + 4/7·49-s + 1.61·55-s + 2.34·59-s + 2.56·61-s + 5/8·64-s − 0.712·71-s − 1.83·76-s − 3.15·79-s − 0.335·80-s + 81-s + 0.635·89-s + 4.92·95-s − 2.01·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4956520453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4956520453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 2856 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 130 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 181 T^{2} + 15312 T^{4} - 181 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 337 T^{2} + 47136 T^{4} - 337 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.51323317728160153252196799477, −11.23905208913192929239537573227, −10.76708718026809165163786206272, −10.55003680888481659010168730285, −10.34868265170857292905220817206, −10.21784533693697580003851458916, −9.572150857601680511869988955596, −9.384390295239582833487040197405, −8.752358483478454298537765325143, −8.561913929967990611253256812393, −8.261424559393711865296506807208, −7.75464132179680785468636652988, −7.72407164836328248452341648958, −7.16134975081704456913463737652, −6.85437210822826851515912330194, −6.76134604263451556055217846691, −6.13306182954689398397886189556, −5.52923801730463138407263999870, −5.42054206796080839394948654022, −4.38975127802683186965272439597, −4.27157937693817625799324775125, −4.12586615836761186344194460914, −3.47984059806640897081261858555, −2.45217711766523776713592093680, −2.09429720371933036108397787319,
2.09429720371933036108397787319, 2.45217711766523776713592093680, 3.47984059806640897081261858555, 4.12586615836761186344194460914, 4.27157937693817625799324775125, 4.38975127802683186965272439597, 5.42054206796080839394948654022, 5.52923801730463138407263999870, 6.13306182954689398397886189556, 6.76134604263451556055217846691, 6.85437210822826851515912330194, 7.16134975081704456913463737652, 7.72407164836328248452341648958, 7.75464132179680785468636652988, 8.261424559393711865296506807208, 8.561913929967990611253256812393, 8.752358483478454298537765325143, 9.384390295239582833487040197405, 9.572150857601680511869988955596, 10.21784533693697580003851458916, 10.34868265170857292905220817206, 10.55003680888481659010168730285, 10.76708718026809165163786206272, 11.23905208913192929239537573227, 11.51323317728160153252196799477
