| L(s) = 1 | + 4·5-s − 2·7-s + 4·17-s + 2·19-s + 4·25-s − 12·29-s + 12·31-s − 8·35-s − 12·37-s + 12·41-s − 12·43-s + 24·47-s + 3·49-s − 4·53-s + 16·59-s + 4·61-s − 24·67-s − 12·73-s + 16·79-s + 16·83-s + 16·85-s + 28·89-s + 8·95-s − 12·97-s − 4·101-s + 16·103-s + 8·107-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s + 0.970·17-s + 0.458·19-s + 4/5·25-s − 2.22·29-s + 2.15·31-s − 1.35·35-s − 1.97·37-s + 1.87·41-s − 1.82·43-s + 3.50·47-s + 3/7·49-s − 0.549·53-s + 2.08·59-s + 0.512·61-s − 2.93·67-s − 1.40·73-s + 1.80·79-s + 1.75·83-s + 1.73·85-s + 2.96·89-s + 0.820·95-s − 1.21·97-s − 0.398·101-s + 1.57·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.450836080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.450836080\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 24 T + 236 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 24 T + 270 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 212 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 28 T + 366 T^{2} - 28 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.76083065100596404967286202438, −7.54694947114195371992072396844, −7.15537255747868464387964204372, −6.78634239099312929708631910917, −6.28853273802384896512902423654, −6.22461875219590100730365318687, −5.78461370774411260812233120190, −5.56213161621931637802152680379, −5.13168951206594420682174749043, −5.10996051493200863459940093253, −4.16427336919708620066028792214, −4.13950735315767457034533560760, −3.39321886145384335730091767706, −3.38850080721243033707546338320, −2.56091492195615597908708683710, −2.53594682515733772089923285746, −1.90861539335780538358564877723, −1.62583553750353325593138593093, −0.982755894106937492270529989039, −0.51139164944939561271452412684,
0.51139164944939561271452412684, 0.982755894106937492270529989039, 1.62583553750353325593138593093, 1.90861539335780538358564877723, 2.53594682515733772089923285746, 2.56091492195615597908708683710, 3.38850080721243033707546338320, 3.39321886145384335730091767706, 4.13950735315767457034533560760, 4.16427336919708620066028792214, 5.10996051493200863459940093253, 5.13168951206594420682174749043, 5.56213161621931637802152680379, 5.78461370774411260812233120190, 6.22461875219590100730365318687, 6.28853273802384896512902423654, 6.78634239099312929708631910917, 7.15537255747868464387964204372, 7.54694947114195371992072396844, 7.76083065100596404967286202438
