| L(s) = 1 | − 4·4-s + 12·16-s − 10·25-s + 8·31-s + 28·49-s − 32·64-s + 56·79-s + 40·100-s + 4·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 112·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 2·25-s + 1.43·31-s + 4·49-s − 4·64-s + 6.30·79-s + 4·100-s + 4/11·121-s − 2.87·124-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 + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 8·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.110915424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110915424\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−8.454277839923177052897084485410, −8.024631287508971036416228629671, −7.77472659363087873887294107987, −7.76241569416037228531766315761, −7.48433071739903033841009052676, −7.17994434650798021870876130076, −6.69464312490420043592651254763, −6.40060107780927609488789468359, −6.28824334314141833793795985233, −5.91383316039994105522590822993, −5.63888434829617977217649930972, −5.40656403214343627806968857069, −5.10082289346785253358422820986, −4.93550850355317429632472861761, −4.59283801964257390722453744422, −4.13296508954357498512058083675, −4.06656873665894839718196062428, −3.71032303271125624701282049567, −3.59317917963346676892765135471, −3.09845057699046422838914145274, −2.48510880608957523542355183570, −2.34041013222452959040818539994, −1.71300875044962520823166294165, −0.950694344106414543605407735459, −0.59134674158970476226223242617,
0.59134674158970476226223242617, 0.950694344106414543605407735459, 1.71300875044962520823166294165, 2.34041013222452959040818539994, 2.48510880608957523542355183570, 3.09845057699046422838914145274, 3.59317917963346676892765135471, 3.71032303271125624701282049567, 4.06656873665894839718196062428, 4.13296508954357498512058083675, 4.59283801964257390722453744422, 4.93550850355317429632472861761, 5.10082289346785253358422820986, 5.40656403214343627806968857069, 5.63888434829617977217649930972, 5.91383316039994105522590822993, 6.28824334314141833793795985233, 6.40060107780927609488789468359, 6.69464312490420043592651254763, 7.17994434650798021870876130076, 7.48433071739903033841009052676, 7.76241569416037228531766315761, 7.77472659363087873887294107987, 8.024631287508971036416228629671, 8.454277839923177052897084485410
