| L(s) = 1 | − 4·3-s + 10·9-s − 8·19-s − 2·25-s − 20·27-s − 8·29-s + 24·31-s + 8·37-s − 24·47-s + 10·49-s + 32·53-s + 32·57-s + 8·75-s + 35·81-s − 8·83-s + 32·87-s − 96·93-s + 16·103-s − 16·109-s − 32·111-s + 16·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s − 1.83·19-s − 2/5·25-s − 3.84·27-s − 1.48·29-s + 4.31·31-s + 1.31·37-s − 3.50·47-s + 10/7·49-s + 4.39·53-s + 4.23·57-s + 0.923·75-s + 35/9·81-s − 0.878·83-s + 3.43·87-s − 9.95·93-s + 1.57·103-s − 1.53·109-s − 3.03·111-s + 1.50·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.491127956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491127956\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 230 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 3590 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 16 T + 164 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2454 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 4842 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 24098 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 17538 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 146 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4134 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 80 T^{2} - 1182 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.82082935684699473034314427872, −6.39310944672633819693792870059, −6.09708195480049410592908980568, −5.98537840721734098104813624009, −5.87090796664935224169919867972, −5.62597329605530382688861509857, −5.55936891904588257717656625734, −5.19943111932846806777641910098, −4.92217483607973195326432067317, −4.50419299566149281052393727950, −4.47099936760462599918936411458, −4.45901262693946976677570613880, −4.29572675139472369904697823984, −3.93398451713396176324779317459, −3.57066070476332587121463519156, −3.21401872148944470730093351986, −3.13432532907619774328530191135, −2.65838867898165098888356674474, −2.21844319417755519943451005565, −2.03417542362833575705406283258, −2.01907496421267323732887874048, −1.29752429742677479077913819932, −1.01665175142953185847810743313, −0.56802810891211087381033273431, −0.47677123588595160222017012881,
0.47677123588595160222017012881, 0.56802810891211087381033273431, 1.01665175142953185847810743313, 1.29752429742677479077913819932, 2.01907496421267323732887874048, 2.03417542362833575705406283258, 2.21844319417755519943451005565, 2.65838867898165098888356674474, 3.13432532907619774328530191135, 3.21401872148944470730093351986, 3.57066070476332587121463519156, 3.93398451713396176324779317459, 4.29572675139472369904697823984, 4.45901262693946976677570613880, 4.47099936760462599918936411458, 4.50419299566149281052393727950, 4.92217483607973195326432067317, 5.19943111932846806777641910098, 5.55936891904588257717656625734, 5.62597329605530382688861509857, 5.87090796664935224169919867972, 5.98537840721734098104813624009, 6.09708195480049410592908980568, 6.39310944672633819693792870059, 6.82082935684699473034314427872
