| L(s) = 1 | + 3-s + 2·5-s − 12·13-s + 2·15-s − 2·17-s − 4·19-s − 4·23-s + 5·25-s − 27-s − 20·29-s + 8·31-s − 6·37-s − 12·39-s + 4·41-s + 8·43-s − 8·47-s − 2·51-s + 10·53-s − 4·57-s − 12·59-s − 2·61-s − 24·65-s + 12·67-s − 4·69-s + 24·71-s − 14·73-s + 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 3.32·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 25-s − 0.192·27-s − 3.71·29-s + 1.43·31-s − 0.986·37-s − 1.92·39-s + 0.624·41-s + 1.21·43-s − 1.16·47-s − 0.280·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s − 2.97·65-s + 1.46·67-s − 0.481·69-s + 2.84·71-s − 1.63·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7699150641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7699150641\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.472349341604462403841358490056, −8.696968124891315927604816933819, −8.625662616132641324420303275209, −7.78996411351314753937867133967, −7.76506429419822287947220711457, −7.15689287777917210912354366575, −7.11228231587564261040551580900, −6.50963129853844191391204897558, −5.96627383423334745831303693689, −5.70692759499162746103138787486, −5.16159610909159105572984952089, −4.69950536536891106020266045636, −4.64847948427257203652237315370, −3.70230190531227872927514265696, −3.61131101873430954589346341482, −2.53675695643368377915193173712, −2.38679290427768111827473991697, −2.22989096279616968007982541764, −1.53735686046447107172648826928, −0.25954707695368090163536175168,
0.25954707695368090163536175168, 1.53735686046447107172648826928, 2.22989096279616968007982541764, 2.38679290427768111827473991697, 2.53675695643368377915193173712, 3.61131101873430954589346341482, 3.70230190531227872927514265696, 4.64847948427257203652237315370, 4.69950536536891106020266045636, 5.16159610909159105572984952089, 5.70692759499162746103138787486, 5.96627383423334745831303693689, 6.50963129853844191391204897558, 7.11228231587564261040551580900, 7.15689287777917210912354366575, 7.76506429419822287947220711457, 7.78996411351314753937867133967, 8.625662616132641324420303275209, 8.696968124891315927604816933819, 9.472349341604462403841358490056
