L(s) = 1 | + 16·17-s + 32·41-s − 16·49-s − 16·73-s − 18·81-s − 8·89-s − 16·97-s − 64·113-s − 4·121-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 + ⋯ |
L(s) = 1 | + 3.88·17-s + 4.99·41-s − 2.28·49-s − 1.87·73-s − 2·81-s − 0.847·89-s − 1.62·97-s − 6.02·113-s − 0.363·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.535056141\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.535056141\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
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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
−6.05615970410450838819862458188, −5.74297752123758488019084904234, −5.65626846175397554527131628310, −5.57019011424309132910613501926, −5.42634051287945923932890079260, −5.41922919194250910766871112110, −4.86486967132369176020490269809, −4.71815099548844107541352548954, −4.43273996139963086316253758054, −4.25454966209769834597051377554, −4.13362607902888671516791593976, −3.83808298121547265194772542997, −3.66799161398898028075613120390, −3.34016698711957625242908283002, −3.13877644325864832603318243123, −2.99033664233518860145287031558, −2.60060082702327761008654131609, −2.54735139689218609257196485996, −2.47817964188556108632166937872, −1.72699189599058183697907838199, −1.39526368606915810095911473862, −1.39271074810046929849735452806, −1.16624033772278006589608708772, −0.75606482757791086251310413829, −0.28510648836256254872603090680,
0.28510648836256254872603090680, 0.75606482757791086251310413829, 1.16624033772278006589608708772, 1.39271074810046929849735452806, 1.39526368606915810095911473862, 1.72699189599058183697907838199, 2.47817964188556108632166937872, 2.54735139689218609257196485996, 2.60060082702327761008654131609, 2.99033664233518860145287031558, 3.13877644325864832603318243123, 3.34016698711957625242908283002, 3.66799161398898028075613120390, 3.83808298121547265194772542997, 4.13362607902888671516791593976, 4.25454966209769834597051377554, 4.43273996139963086316253758054, 4.71815099548844107541352548954, 4.86486967132369176020490269809, 5.41922919194250910766871112110, 5.42634051287945923932890079260, 5.57019011424309132910613501926, 5.65626846175397554527131628310, 5.74297752123758488019084904234, 6.05615970410450838819862458188