| L(s) = 1 | − 4·7-s − 14·25-s + 20·37-s − 20·43-s − 2·49-s − 8·67-s + 52·79-s − 28·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 56·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 2.79·25-s + 3.28·37-s − 3.04·43-s − 2/7·49-s − 0.977·67-s + 5.85·79-s − 2.68·109-s + 3.63·121-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 + 3.07·169-s + 0.0760·173-s + 4.23·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610633978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610633978\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 163 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.21397935896695002107825542070, −6.13996899913554701368300562726, −5.73531807293022851674778658168, −5.71489816820015842887008912033, −5.37230710686915281675145381744, −5.06758980826642576927217905432, −5.04955768075865872459875128840, −4.86604173409420923556668248055, −4.39397653356963753801259640463, −4.24918047807647133560683929828, −4.16310535481772953083875700005, −3.85211206974430823711303568972, −3.55708490797448211037227225228, −3.48785239393292837442239817350, −3.19701259060450541425572749375, −3.11923114137449066387456638536, −2.64485509824527501329535462750, −2.53391645658292052249275060187, −2.22086211762820082792330300084, −1.95575135540242145309739374218, −1.71976668654173784339913194175, −1.42506369053523215070259667415, −0.946872340514202530897716208534, −0.50176122247856253311475649149, −0.29673155227346828015419320557,
0.29673155227346828015419320557, 0.50176122247856253311475649149, 0.946872340514202530897716208534, 1.42506369053523215070259667415, 1.71976668654173784339913194175, 1.95575135540242145309739374218, 2.22086211762820082792330300084, 2.53391645658292052249275060187, 2.64485509824527501329535462750, 3.11923114137449066387456638536, 3.19701259060450541425572749375, 3.48785239393292837442239817350, 3.55708490797448211037227225228, 3.85211206974430823711303568972, 4.16310535481772953083875700005, 4.24918047807647133560683929828, 4.39397653356963753801259640463, 4.86604173409420923556668248055, 5.04955768075865872459875128840, 5.06758980826642576927217905432, 5.37230710686915281675145381744, 5.71489816820015842887008912033, 5.73531807293022851674778658168, 6.13996899913554701368300562726, 6.21397935896695002107825542070
