| L(s) = 1 | + 4·5-s + 4·11-s + 12·17-s − 8·19-s + 4·23-s + 4·25-s + 8·37-s + 12·41-s + 8·47-s + 4·53-s + 16·55-s − 8·61-s + 4·71-s − 24·73-s + 16·79-s − 8·83-s + 48·85-s + 20·89-s − 32·95-s − 8·97-s + 4·101-s − 20·107-s + 16·109-s + 20·113-s + 16·115-s − 2·121-s − 12·125-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.20·11-s + 2.91·17-s − 1.83·19-s + 0.834·23-s + 4/5·25-s + 1.31·37-s + 1.87·41-s + 1.16·47-s + 0.549·53-s + 2.15·55-s − 1.02·61-s + 0.474·71-s − 2.80·73-s + 1.80·79-s − 0.878·83-s + 5.20·85-s + 2.11·89-s − 3.28·95-s − 0.812·97-s + 0.398·101-s − 1.93·107-s + 1.53·109-s + 1.88·113-s + 1.49·115-s − 0.181·121-s − 1.07·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.992466204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.992466204\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 4 p T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 24 T + 272 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.730875012668093835389247836763, −8.567815856676604029808553655002, −8.007538356040298577271433961624, −7.51417405047175509523058001380, −7.38375461724293144622470931010, −6.87186931023361883495282996213, −6.24680512309991632005532449435, −6.07713557908784820834491545107, −5.85372425671583464457774672447, −5.63676407497423712376550962060, −4.99852055127935549907253676068, −4.60163990902515920364010995251, −4.03843381985792987666362365924, −3.82657241877912353514283211245, −2.97118326152500468249545983317, −2.92304816601555277403251884257, −2.00402896934612266725331446311, −1.94125724032394113444315883842, −1.07659706948853737682306136002, −0.881643203536167302797159610420,
0.881643203536167302797159610420, 1.07659706948853737682306136002, 1.94125724032394113444315883842, 2.00402896934612266725331446311, 2.92304816601555277403251884257, 2.97118326152500468249545983317, 3.82657241877912353514283211245, 4.03843381985792987666362365924, 4.60163990902515920364010995251, 4.99852055127935549907253676068, 5.63676407497423712376550962060, 5.85372425671583464457774672447, 6.07713557908784820834491545107, 6.24680512309991632005532449435, 6.87186931023361883495282996213, 7.38375461724293144622470931010, 7.51417405047175509523058001380, 8.007538356040298577271433961624, 8.567815856676604029808553655002, 8.730875012668093835389247836763
