| L(s) = 1 | − 2·4-s + 4·9-s + 4·11-s + 3·16-s − 20·19-s + 12·29-s + 8·31-s − 8·36-s + 12·41-s − 8·44-s − 2·49-s + 8·61-s − 4·64-s + 24·71-s + 40·76-s + 28·79-s + 6·81-s + 16·99-s + 24·101-s + 28·109-s − 24·116-s + 10·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s + 4/3·9-s + 1.20·11-s + 3/4·16-s − 4.58·19-s + 2.22·29-s + 1.43·31-s − 4/3·36-s + 1.87·41-s − 1.20·44-s − 2/7·49-s + 1.02·61-s − 1/2·64-s + 2.84·71-s + 4.58·76-s + 3.15·79-s + 2/3·81-s + 1.60·99-s + 2.38·101-s + 2.68·109-s − 2.22·116-s + 0.909·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.685954982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.685954982\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 7626 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 5130 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4230 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 180 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 1290 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 20346 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.02394424276789619713875093028, −5.96905765013831786636805037301, −5.73855956595184080468036570108, −5.12706148188751085476362854655, −5.11170848095506488781952743094, −4.88535195895370744197774625630, −4.87925904715799489005187266624, −4.46878826686280481761640161156, −4.31900031461649680438315607664, −4.15193840301951543148447999558, −4.14599854660980328190164171188, −3.86065932318026988272113766546, −3.79786543890403608764078901730, −3.38668970942343803645993342607, −3.03712525212981341941203744998, −2.93300624718859933761900532654, −2.59252605085204713946045447605, −2.19029512059966163251576509320, −1.99836628076775742004816954794, −1.98799247020096320542397459294, −1.77716866454394280728398949340, −1.02984674606785596243924365371, −0.865374769396671347688845271207, −0.792041507876167824447267010982, −0.37272446864694396930898570592,
0.37272446864694396930898570592, 0.792041507876167824447267010982, 0.865374769396671347688845271207, 1.02984674606785596243924365371, 1.77716866454394280728398949340, 1.98799247020096320542397459294, 1.99836628076775742004816954794, 2.19029512059966163251576509320, 2.59252605085204713946045447605, 2.93300624718859933761900532654, 3.03712525212981341941203744998, 3.38668970942343803645993342607, 3.79786543890403608764078901730, 3.86065932318026988272113766546, 4.14599854660980328190164171188, 4.15193840301951543148447999558, 4.31900031461649680438315607664, 4.46878826686280481761640161156, 4.87925904715799489005187266624, 4.88535195895370744197774625630, 5.11170848095506488781952743094, 5.12706148188751085476362854655, 5.73855956595184080468036570108, 5.96905765013831786636805037301, 6.02394424276789619713875093028
