| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 6·11-s − 3·13-s + 3·17-s + 5·19-s + 3·21-s − 6·23-s − 5·25-s + 9·27-s − 3·29-s − 2·31-s + 18·33-s − 7·37-s − 9·39-s − 3·41-s − 3·43-s + 18·47-s − 6·49-s + 9·51-s + 9·53-s + 15·57-s + 30·59-s + 6·63-s − 18·69-s + 3·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 1.80·11-s − 0.832·13-s + 0.727·17-s + 1.14·19-s + 0.654·21-s − 1.25·23-s − 25-s + 1.73·27-s − 0.557·29-s − 0.359·31-s + 3.13·33-s − 1.15·37-s − 1.44·39-s − 0.468·41-s − 0.457·43-s + 2.62·47-s − 6/7·49-s + 1.26·51-s + 1.23·53-s + 1.98·57-s + 3.90·59-s + 0.755·63-s − 2.16·69-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.035844409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.035844409\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 155 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−9.837147548740087360524857288925, −9.724511980708443519815528802470, −9.425171244275730096917465766313, −8.821488985283046628773388187527, −8.623244163171457247585221814413, −8.132590810976590708295117695620, −7.73371322268840453574308951633, −7.32063871918843431211814253998, −6.95969319067565421437770103243, −6.65791770779633399738343930165, −5.65013247888171610059278431792, −5.58985726301466298688621424048, −4.86595464522402419086456869466, −4.03601918940879559176678853399, −3.81101404945237639188635065524, −3.66367562909593689065816477613, −2.79519533687752134454813965518, −2.17742987794193203769958104679, −1.76862279691920699171063195345, −1.00566263904630957776191389759,
1.00566263904630957776191389759, 1.76862279691920699171063195345, 2.17742987794193203769958104679, 2.79519533687752134454813965518, 3.66367562909593689065816477613, 3.81101404945237639188635065524, 4.03601918940879559176678853399, 4.86595464522402419086456869466, 5.58985726301466298688621424048, 5.65013247888171610059278431792, 6.65791770779633399738343930165, 6.95969319067565421437770103243, 7.32063871918843431211814253998, 7.73371322268840453574308951633, 8.132590810976590708295117695620, 8.623244163171457247585221814413, 8.821488985283046628773388187527, 9.425171244275730096917465766313, 9.724511980708443519815528802470, 9.837147548740087360524857288925
