| L(s) = 1 | − 64·9-s − 112·17-s − 412·25-s − 840·41-s − 1.35e3·49-s − 2.91e3·73-s + 1.61e3·81-s − 2.46e3·89-s − 3.92e3·97-s − 936·113-s − 3.16e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 7.16e3·153-s + 157-s + 163-s + 167-s − 1.66e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2.37·9-s − 1.59·17-s − 3.29·25-s − 3.19·41-s − 3.95·49-s − 4.66·73-s + 2.21·81-s − 2.93·89-s − 4.10·97-s − 0.779·113-s − 2.38·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 3.78·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.755·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 206 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 678 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 144 p T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 830 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 7360 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 22022 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 46622 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 58014 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 73330 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 132064 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 99682 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 36878 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 244240 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 232158 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 574576 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 56870 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 728 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 721474 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 92144 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 616 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 980 T + p^{3} T^{2} )^{4} \) |
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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
−8.206811340103113944649788048770, −7.56789249428859113614500463589, −7.56447601071542573066981619491, −7.12034309317825504055100458603, −7.08412033326346858638307212042, −6.43192969203266994758320556526, −6.31953479652496821546015462886, −6.26837039970570805505145263405, −6.25260368977466191378799159937, −5.47455148194537952489063053359, −5.37443538431616080242438163628, −5.33686502256184595320229264809, −5.24639297979583717020926904174, −4.54567351352493162608377144424, −4.22952994048067119207463751833, −4.15267623167561185260022492295, −3.97454989810089844496000913556, −3.28161852109922866586601931362, −3.12574130966874351380666606088, −2.96790886532333656861761923507, −2.71997490213699087637395309333, −2.15280051588321523272294427406, −1.88834973651697442331045199054, −1.53159189538271055551244424273, −1.35662916567376012586030605043, 0, 0, 0, 0,
1.35662916567376012586030605043, 1.53159189538271055551244424273, 1.88834973651697442331045199054, 2.15280051588321523272294427406, 2.71997490213699087637395309333, 2.96790886532333656861761923507, 3.12574130966874351380666606088, 3.28161852109922866586601931362, 3.97454989810089844496000913556, 4.15267623167561185260022492295, 4.22952994048067119207463751833, 4.54567351352493162608377144424, 5.24639297979583717020926904174, 5.33686502256184595320229264809, 5.37443538431616080242438163628, 5.47455148194537952489063053359, 6.25260368977466191378799159937, 6.26837039970570805505145263405, 6.31953479652496821546015462886, 6.43192969203266994758320556526, 7.08412033326346858638307212042, 7.12034309317825504055100458603, 7.56447601071542573066981619491, 7.56789249428859113614500463589, 8.206811340103113944649788048770
