| L(s) = 1 | + 8·11-s − 8·19-s + 12·29-s − 16·31-s + 12·41-s + 14·49-s − 8·59-s − 4·61-s + 16·71-s − 16·79-s − 12·89-s + 36·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.83·19-s + 2.22·29-s − 2.87·31-s + 1.87·41-s + 2·49-s − 1.04·59-s − 0.512·61-s + 1.89·71-s − 1.80·79-s − 1.27·89-s + 3.58·101-s + 0.383·109-s + 2.36·121-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.031204542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.031204542\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.880235549524611474629892471213, −8.649792764376536655361070287795, −7.901922215824919972883868166850, −7.68336374031468174486449227595, −7.14769381142733388377539407613, −6.84837289967856133824018793257, −6.49298552746777437529804781851, −6.25023235939032560187135724992, −5.78439418326447439891819911254, −5.51593197042013070059121188915, −4.79624636091922179793487598575, −4.38693585929860151054920179402, −4.04987425194175295140642404569, −3.93082385162742754470406013565, −3.27685745960515832228583320113, −2.79828932804449657071578923926, −2.08960821207701551930732862496, −1.82982338782789702157272157662, −1.14152182841781339007678171368, −0.57218158202813785377166073906,
0.57218158202813785377166073906, 1.14152182841781339007678171368, 1.82982338782789702157272157662, 2.08960821207701551930732862496, 2.79828932804449657071578923926, 3.27685745960515832228583320113, 3.93082385162742754470406013565, 4.04987425194175295140642404569, 4.38693585929860151054920179402, 4.79624636091922179793487598575, 5.51593197042013070059121188915, 5.78439418326447439891819911254, 6.25023235939032560187135724992, 6.49298552746777437529804781851, 6.84837289967856133824018793257, 7.14769381142733388377539407613, 7.68336374031468174486449227595, 7.901922215824919972883868166850, 8.649792764376536655361070287795, 8.880235549524611474629892471213
