| L(s) = 1 | + 5·9-s − 16·13-s − 2·25-s − 4·37-s + 12·61-s + 28·73-s + 16·81-s + 32·97-s + 20·109-s − 80·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 4.43·13-s − 2/5·25-s − 0.657·37-s + 1.53·61-s + 3.27·73-s + 16/9·81-s + 3.24·97-s + 1.91·109-s − 7.39·117-s − 2·121-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 + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.018619873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.018619873\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 107 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 167 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−6.40002027221073859349949252581, −6.38130678507351713748243794154, −6.04841372145232596017626697262, −5.59953796382860905098672878526, −5.36370043080175001735708744818, −5.20264489347812509052820192454, −5.11300003502230271010769247357, −4.99688988708014104975088302908, −4.80404672906441903959002871156, −4.37495988091423507285528000430, −4.29162284538368623740324640065, −4.25812783298571084350987260458, −3.71973113080277983151233657025, −3.67325216289455862058976769645, −3.38123783756840564763076096595, −2.97252362417964098296084082983, −2.72267718293821662937868102610, −2.59473417873849938688599457430, −2.22570325336136179062630474824, −2.02447006964472145258521515744, −1.77135205231088166297709692721, −1.74781712990058556735330510964, −0.929981671462054324155053037832, −0.59744382209633611690000424288, −0.40281135957156093806881923988,
0.40281135957156093806881923988, 0.59744382209633611690000424288, 0.929981671462054324155053037832, 1.74781712990058556735330510964, 1.77135205231088166297709692721, 2.02447006964472145258521515744, 2.22570325336136179062630474824, 2.59473417873849938688599457430, 2.72267718293821662937868102610, 2.97252362417964098296084082983, 3.38123783756840564763076096595, 3.67325216289455862058976769645, 3.71973113080277983151233657025, 4.25812783298571084350987260458, 4.29162284538368623740324640065, 4.37495988091423507285528000430, 4.80404672906441903959002871156, 4.99688988708014104975088302908, 5.11300003502230271010769247357, 5.20264489347812509052820192454, 5.36370043080175001735708744818, 5.59953796382860905098672878526, 6.04841372145232596017626697262, 6.38130678507351713748243794154, 6.40002027221073859349949252581
