| L(s) = 1 | − 2·4-s + 8·7-s + 3·16-s + 4·17-s − 12·19-s + 8·25-s − 16·28-s − 24·29-s − 8·31-s − 16·47-s + 32·49-s − 16·59-s + 40·61-s − 4·64-s − 12·67-s − 8·68-s − 24·71-s + 24·73-s + 24·76-s + 16·79-s + 14·81-s + 8·83-s + 24·89-s − 28·97-s − 16·100-s − 8·103-s − 8·107-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s + 3/4·16-s + 0.970·17-s − 2.75·19-s + 8/5·25-s − 3.02·28-s − 4.45·29-s − 1.43·31-s − 2.33·47-s + 32/7·49-s − 2.08·59-s + 5.12·61-s − 1/2·64-s − 1.46·67-s − 0.970·68-s − 2.84·71-s + 2.80·73-s + 2.75·76-s + 1.80·79-s + 14/9·81-s + 0.878·83-s + 2.54·89-s − 2.84·97-s − 8/5·100-s − 0.788·103-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737959846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737959846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 16 p T^{3} + p^{3} T^{4} - 16 p^{2} T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 379 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 252 T^{3} + 878 T^{4} + 252 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1714 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2400 T^{3} + 14959 T^{4} + 2400 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 240 T^{3} + 1799 T^{4} + 240 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 688 T^{3} + 3682 T^{4} + 688 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3791 T^{4} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1200 T^{3} + 10994 T^{4} + 1200 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 10400 T^{3} + 95599 T^{4} - 10400 p T^{5} + 800 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 588 T^{3} + 4478 T^{4} + 588 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 752 T^{3} + 3394 T^{4} - 752 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 600 T^{3} + 11186 T^{4} - 600 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 3768 T^{3} + 44674 T^{4} - 3768 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
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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
−8.343609756502396127882474464457, −8.217866139929134453537099961520, −7.72546834366340449129355886381, −7.59437514731611160660224904917, −7.49529724677545424973747618310, −7.13203502697777617605783713840, −6.83667352161600449688762203691, −6.53112068758936500721581705491, −6.00126260272100583972998938750, −5.93358108964831542927351893279, −5.69087031663801276301996249390, −5.00421882926259100207422635366, −5.00273884604498735443883274992, −4.99533050786771309060333121418, −4.94017374713547160644424335116, −4.06745856220453554338825131296, −4.01915241725879251382405912823, −3.86903106372955036812570944069, −3.51354717052311284685673312934, −3.02904831970757363393540958919, −2.26692685529396963735913351243, −2.02731338382684470744773368679, −1.74954329926926999374904816106, −1.50391912492268737861419722580, −0.54072111582990698971622770705,
0.54072111582990698971622770705, 1.50391912492268737861419722580, 1.74954329926926999374904816106, 2.02731338382684470744773368679, 2.26692685529396963735913351243, 3.02904831970757363393540958919, 3.51354717052311284685673312934, 3.86903106372955036812570944069, 4.01915241725879251382405912823, 4.06745856220453554338825131296, 4.94017374713547160644424335116, 4.99533050786771309060333121418, 5.00273884604498735443883274992, 5.00421882926259100207422635366, 5.69087031663801276301996249390, 5.93358108964831542927351893279, 6.00126260272100583972998938750, 6.53112068758936500721581705491, 6.83667352161600449688762203691, 7.13203502697777617605783713840, 7.49529724677545424973747618310, 7.59437514731611160660224904917, 7.72546834366340449129355886381, 8.217866139929134453537099961520, 8.343609756502396127882474464457
