| L(s) = 1 | − 4·4-s + 12·16-s − 20·25-s + 8·43-s + 14·49-s − 32·64-s − 40·67-s + 80·100-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s − 32·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 56·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 4·25-s + 1.21·43-s + 2·49-s − 4·64-s − 4.88·67-s + 8·100-s − 4·121-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 + 4/13·169-s − 2.43·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1300207596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1300207596\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−7.82542503405906086005316133827, −7.70847525422549412785135462928, −7.49009539013041866720969105480, −7.32435750461841954403926244833, −7.21918409230934678474744023169, −6.45382725006597471321306767225, −6.31651390568438319190576419983, −6.07211144745850299516812234115, −5.83755660888096284262870974688, −5.71887489575259466224404756806, −5.35245870528907269225975476162, −5.21959637345286850389778043082, −4.79995957462417333315078955086, −4.53261572760928652824758187709, −4.16064833851770811675492977775, −4.11110637263828032278611242875, −3.78271851312431008491723456974, −3.70548012195453083070796034945, −3.17967664538115002665494991319, −2.76693032252624515375797983324, −2.48310195227880672755422352283, −1.93993926995773831038029751614, −1.48453636019026352839449672475, −1.09404301299056981693825080023, −0.13996510914528852140124123685,
0.13996510914528852140124123685, 1.09404301299056981693825080023, 1.48453636019026352839449672475, 1.93993926995773831038029751614, 2.48310195227880672755422352283, 2.76693032252624515375797983324, 3.17967664538115002665494991319, 3.70548012195453083070796034945, 3.78271851312431008491723456974, 4.11110637263828032278611242875, 4.16064833851770811675492977775, 4.53261572760928652824758187709, 4.79995957462417333315078955086, 5.21959637345286850389778043082, 5.35245870528907269225975476162, 5.71887489575259466224404756806, 5.83755660888096284262870974688, 6.07211144745850299516812234115, 6.31651390568438319190576419983, 6.45382725006597471321306767225, 7.21918409230934678474744023169, 7.32435750461841954403926244833, 7.49009539013041866720969105480, 7.70847525422549412785135462928, 7.82542503405906086005316133827
