| L(s) = 1 | − 4·5-s − 8·7-s − 12·17-s − 8·23-s + 8·25-s + 4·29-s − 8·31-s + 32·35-s − 12·37-s − 20·41-s − 16·47-s + 32·49-s + 20·61-s − 16·67-s − 24·71-s + 4·73-s + 24·79-s − 81-s + 48·85-s − 4·97-s + 32·101-s − 48·103-s − 16·107-s − 28·109-s + 36·113-s + 32·115-s + 96·119-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 3.02·7-s − 2.91·17-s − 1.66·23-s + 8/5·25-s + 0.742·29-s − 1.43·31-s + 5.40·35-s − 1.97·37-s − 3.12·41-s − 2.33·47-s + 32/7·49-s + 2.56·61-s − 1.95·67-s − 2.84·71-s + 0.468·73-s + 2.70·79-s − 1/9·81-s + 5.20·85-s − 0.406·97-s + 3.18·101-s − 4.72·103-s − 1.54·107-s − 2.68·109-s + 3.38·113-s + 2.98·115-s + 8.80·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08884511085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08884511085\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 40 T^{3} - 734 T^{4} - 40 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 108 T^{3} + 1454 T^{4} - 108 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 280 T^{3} + 2434 T^{4} + 280 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 612 T^{3} + 5006 T^{4} + 612 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1500 T^{3} + 10094 T^{4} + 1500 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3254 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1500 T^{3} + 11054 T^{4} - 1500 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 3336 T^{3} + 34018 T^{4} + 3336 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 44 T^{3} - 3602 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2760 T^{3} + 25154 T^{4} - 2760 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 15254 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 180 T^{3} - 16114 T^{4} - 180 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.21165716636773738574134666867, −6.91082377791000376815853331318, −6.87392082324596991184773209405, −6.75191365650503789327786431717, −6.58442141055960776126801332659, −6.38666335690508639466005862691, −6.10599262942241205808333331102, −5.69786293440183547908808473416, −5.63886759728607849511571077480, −5.13931571196666095412858613488, −4.97660732601780130336166098008, −4.69673889535566963949280330929, −4.25596395852176871910506792797, −4.09946727916419022008937505802, −4.07668047594463137237304795054, −3.44288617668749250871361186093, −3.36522509732867451630974552991, −3.35083300859225708871262387172, −3.09835415340347828039694673899, −2.39000333384026040715742061308, −2.37192143888349259971770358756, −1.79405591980708979705655255744, −1.55045624520538736597856146767, −0.29169275728891520489584279049, −0.23492026500528177265833305448,
0.23492026500528177265833305448, 0.29169275728891520489584279049, 1.55045624520538736597856146767, 1.79405591980708979705655255744, 2.37192143888349259971770358756, 2.39000333384026040715742061308, 3.09835415340347828039694673899, 3.35083300859225708871262387172, 3.36522509732867451630974552991, 3.44288617668749250871361186093, 4.07668047594463137237304795054, 4.09946727916419022008937505802, 4.25596395852176871910506792797, 4.69673889535566963949280330929, 4.97660732601780130336166098008, 5.13931571196666095412858613488, 5.63886759728607849511571077480, 5.69786293440183547908808473416, 6.10599262942241205808333331102, 6.38666335690508639466005862691, 6.58442141055960776126801332659, 6.75191365650503789327786431717, 6.87392082324596991184773209405, 6.91082377791000376815853331318, 7.21165716636773738574134666867
