L(s) = 1 | + 4·9-s − 8·49-s − 32·61-s + 7·81-s − 64·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | + 4/3·9-s − 8/7·49-s − 4.09·61-s + 7/9·81-s − 6.13·109-s − 4·121-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 − 4·169-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7774830453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7774830453\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−6.92509261156463247815173193532, −6.65435438878396866518088488243, −6.58856874111609122198008897354, −6.36182810065420050000545529189, −6.08731984419186035290178581779, −6.03634019887939705435999547990, −5.60907769289998755106605175634, −5.19745834807429971860895813244, −5.19478310939508148058502465999, −5.05688802381976880694073409835, −4.69008387541025363264653674050, −4.27312406216852418573863544590, −4.24921464296036442236719590495, −4.15542377000874807303169106813, −3.63487922013968776080021748011, −3.52403684896991389277666314904, −3.24408181734925928681211336652, −2.68445029283898720053637630623, −2.64654296774814580417924265032, −2.52409864054875588747354988307, −1.72726608917248872219843368270, −1.55974140955655215379962410735, −1.46242598362116845334824100896, −1.02500644120511307670571005275, −0.18175536164629201967457014113,
0.18175536164629201967457014113, 1.02500644120511307670571005275, 1.46242598362116845334824100896, 1.55974140955655215379962410735, 1.72726608917248872219843368270, 2.52409864054875588747354988307, 2.64654296774814580417924265032, 2.68445029283898720053637630623, 3.24408181734925928681211336652, 3.52403684896991389277666314904, 3.63487922013968776080021748011, 4.15542377000874807303169106813, 4.24921464296036442236719590495, 4.27312406216852418573863544590, 4.69008387541025363264653674050, 5.05688802381976880694073409835, 5.19478310939508148058502465999, 5.19745834807429971860895813244, 5.60907769289998755106605175634, 6.03634019887939705435999547990, 6.08731984419186035290178581779, 6.36182810065420050000545529189, 6.58856874111609122198008897354, 6.65435438878396866518088488243, 6.92509261156463247815173193532