L(s) = 1 | − 4·13-s + 14·17-s − 10·23-s − 25-s + 20·29-s − 20·43-s − 11·49-s − 6·53-s − 6·61-s + 22·79-s + 20·101-s − 32·103-s − 6·107-s − 12·113-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 1.10·13-s + 3.39·17-s − 2.08·23-s − 1/5·25-s + 3.71·29-s − 3.04·43-s − 1.57·49-s − 0.824·53-s − 0.768·61-s + 2.47·79-s + 1.99·101-s − 3.15·103-s − 0.580·107-s − 1.12·113-s + 1.18·121-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 + 3/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.089506923\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.089506923\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 129 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.432309939435422914674040208681, −8.492441588334282715469981505705, −8.354477193559563192615352072039, −7.923022058846467000296861489394, −7.909517179003454954694443801062, −7.37319795215400871023469472377, −6.79249690913896512162748077361, −6.33687979946903108045394213671, −6.26974604145328352693822555295, −5.56997482234549292974850643832, −5.20748452789313535586009164314, −4.82209976237409628280556190302, −4.56458745122432089651489905648, −3.77991099481154223672250760835, −3.36406648445537590876230411688, −3.04624712753436455867114710652, −2.56680637417157147439559997875, −1.73965112852503995105138626493, −1.33987262118126938726585967187, −0.51610018215352404524271297822,
0.51610018215352404524271297822, 1.33987262118126938726585967187, 1.73965112852503995105138626493, 2.56680637417157147439559997875, 3.04624712753436455867114710652, 3.36406648445537590876230411688, 3.77991099481154223672250760835, 4.56458745122432089651489905648, 4.82209976237409628280556190302, 5.20748452789313535586009164314, 5.56997482234549292974850643832, 6.26974604145328352693822555295, 6.33687979946903108045394213671, 6.79249690913896512162748077361, 7.37319795215400871023469472377, 7.909517179003454954694443801062, 7.923022058846467000296861489394, 8.354477193559563192615352072039, 8.492441588334282715469981505705, 9.432309939435422914674040208681