| L(s) = 1 | − 2·4-s + 3·16-s + 4·25-s − 20·37-s + 4·43-s − 4·64-s − 52·67-s − 28·79-s − 8·100-s − 44·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 8·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s + 4/5·25-s − 3.28·37-s + 0.609·43-s − 1/2·64-s − 6.35·67-s − 3.15·79-s − 4/5·100-s − 4.21·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.28·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s − 0.609·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3456392054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3456392054\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 191 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.39700570976664773637345725460, −5.85401882131939502689448848630, −5.81920775758457394669622206698, −5.79775956898729590476588660660, −5.41738614774005777433898050750, −5.16312677064791040863063643269, −5.06978698900070101025934260169, −4.95467483256283754615820370696, −4.56135623003527393968994455505, −4.26914525237756257470380061858, −4.18915930416419034521806935969, −4.14789006328909177337762873711, −3.88345623291407060080117300649, −3.42590930339096614781817055359, −3.13316084887082787238013189480, −3.05749750129489386307356277901, −2.85459611897854463517318933179, −2.83476177083578911548840288168, −2.10158385709823255651181311462, −1.95296863211110030335621585762, −1.58968893226636122928646063207, −1.39723544372457057026105391958, −1.19528883394812666389993654710, −0.54385566448260195604611797390, −0.12958975186300106178134200137,
0.12958975186300106178134200137, 0.54385566448260195604611797390, 1.19528883394812666389993654710, 1.39723544372457057026105391958, 1.58968893226636122928646063207, 1.95296863211110030335621585762, 2.10158385709823255651181311462, 2.83476177083578911548840288168, 2.85459611897854463517318933179, 3.05749750129489386307356277901, 3.13316084887082787238013189480, 3.42590930339096614781817055359, 3.88345623291407060080117300649, 4.14789006328909177337762873711, 4.18915930416419034521806935969, 4.26914525237756257470380061858, 4.56135623003527393968994455505, 4.95467483256283754615820370696, 5.06978698900070101025934260169, 5.16312677064791040863063643269, 5.41738614774005777433898050750, 5.79775956898729590476588660660, 5.81920775758457394669622206698, 5.85401882131939502689448848630, 6.39700570976664773637345725460
