| L(s) = 1 | − 4·13-s + 14·25-s + 16·37-s + 10·49-s + 28·61-s + 16·73-s + 4·97-s − 16·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.10·13-s + 14/5·25-s + 2.63·37-s + 10/7·49-s + 3.58·61-s + 1.87·73-s + 0.406·97-s − 1.53·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.838877859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.838877859\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 166 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
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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
−5.83803446249892652618841646143, −5.62392988546889275522546705698, −5.28285344404272357640211333980, −5.19116596021185894088894066495, −4.89685604241701962113814737178, −4.85832008345101056193200477941, −4.77685121394543275203068889712, −4.46688717438742247857060477741, −4.31423161643641140724429963380, −3.86516692939501432175664717216, −3.76438931078635866255006621613, −3.76071828344010749856237860631, −3.49639214701660256852741795928, −3.06033802096108669456280492844, −2.92620695118381712624187567453, −2.61496924091478691355906014639, −2.42940190010054195981045922369, −2.41677449623091275869148967348, −2.23575348587454744917422663249, −1.84804952332720994998663885801, −1.30318701461698041904794590141, −1.17232038344762275012936198836, −0.904401654302912562778234477612, −0.73058589671635108842815333058, −0.26822038575757211468173403870,
0.26822038575757211468173403870, 0.73058589671635108842815333058, 0.904401654302912562778234477612, 1.17232038344762275012936198836, 1.30318701461698041904794590141, 1.84804952332720994998663885801, 2.23575348587454744917422663249, 2.41677449623091275869148967348, 2.42940190010054195981045922369, 2.61496924091478691355906014639, 2.92620695118381712624187567453, 3.06033802096108669456280492844, 3.49639214701660256852741795928, 3.76071828344010749856237860631, 3.76438931078635866255006621613, 3.86516692939501432175664717216, 4.31423161643641140724429963380, 4.46688717438742247857060477741, 4.77685121394543275203068889712, 4.85832008345101056193200477941, 4.89685604241701962113814737178, 5.19116596021185894088894066495, 5.28285344404272357640211333980, 5.62392988546889275522546705698, 5.83803446249892652618841646143
