| L(s) = 1 | + 2·3-s + 4·5-s + 6·7-s + 2·9-s + 8·15-s + 12·21-s + 10·25-s + 6·27-s + 24·35-s + 32·41-s + 20·43-s + 8·45-s − 4·47-s + 18·49-s + 12·63-s − 12·67-s + 20·75-s − 24·79-s + 11·81-s − 20·83-s + 32·89-s − 8·101-s + 48·105-s + 24·109-s + 16·121-s + 64·123-s + 20·125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 2.26·7-s + 2/3·9-s + 2.06·15-s + 2.61·21-s + 2·25-s + 1.15·27-s + 4.05·35-s + 4.99·41-s + 3.04·43-s + 1.19·45-s − 0.583·47-s + 18/7·49-s + 1.51·63-s − 1.46·67-s + 2.30·75-s − 2.70·79-s + 11/9·81-s − 2.19·83-s + 3.39·89-s − 0.796·101-s + 4.68·105-s + 2.29·109-s + 1.45·121-s + 5.77·123-s + 1.78·125-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
| \(L(1)\) |
\(\approx\) |
\(21.73384810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.73384810\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 286 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5038 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_4$ | \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 7342 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 9838 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 17918 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.67871807071117226569253296551, −6.27870688880799718606298737805, −6.08748149738753223274716933103, −6.03048812070819602853458345429, −5.89684362243154577615975544855, −5.55387481853556264941821087030, −5.41109133899810486251578748803, −5.25402116331487917785964787108, −4.74577941876334570749897244674, −4.68961519967663045300257212291, −4.43619079394787632133460558110, −4.27821513789234498385610297349, −4.24457801198641321980019397214, −3.76477662353238744186008108730, −3.51017889261060928663779038551, −2.95292182682839107690267825389, −2.90675965730011317495914818087, −2.59564926289843490116500942374, −2.50096369320823543103859402749, −2.15999839400117362221158975062, −1.86825343897392893228175158672, −1.70612688346468317981159037595, −1.25920285030274309986933434344, −0.871798362609212949307305579397, −0.851305409811585209475125479449,
0.851305409811585209475125479449, 0.871798362609212949307305579397, 1.25920285030274309986933434344, 1.70612688346468317981159037595, 1.86825343897392893228175158672, 2.15999839400117362221158975062, 2.50096369320823543103859402749, 2.59564926289843490116500942374, 2.90675965730011317495914818087, 2.95292182682839107690267825389, 3.51017889261060928663779038551, 3.76477662353238744186008108730, 4.24457801198641321980019397214, 4.27821513789234498385610297349, 4.43619079394787632133460558110, 4.68961519967663045300257212291, 4.74577941876334570749897244674, 5.25402116331487917785964787108, 5.41109133899810486251578748803, 5.55387481853556264941821087030, 5.89684362243154577615975544855, 6.03048812070819602853458345429, 6.08748149738753223274716933103, 6.27870688880799718606298737805, 6.67871807071117226569253296551
