| L(s) = 1 | − 2·3-s − 9-s − 4·11-s + 4·13-s − 4·17-s + 14·23-s + 6·27-s − 10·29-s + 4·31-s + 8·33-s − 8·39-s − 6·41-s − 6·43-s + 12·47-s + 8·51-s + 8·59-s − 2·61-s + 6·67-s − 28·69-s − 24·71-s − 4·73-s − 8·79-s − 4·81-s − 18·83-s + 20·87-s + 22·89-s − 8·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.970·17-s + 2.91·23-s + 1.15·27-s − 1.85·29-s + 0.718·31-s + 1.39·33-s − 1.28·39-s − 0.937·41-s − 0.914·43-s + 1.75·47-s + 1.12·51-s + 1.04·59-s − 0.256·61-s + 0.733·67-s − 3.37·69-s − 2.84·71-s − 0.468·73-s − 0.900·79-s − 4/9·81-s − 1.97·83-s + 2.14·87-s + 2.33·89-s − 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.461659133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.461659133\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14 T + 93 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T - 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 91 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 229 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 22 T + 3 p T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.52945608121675176218474087850, −7.52581392362316006552579383772, −7.10949935423580399176141247337, −6.88976013789666891737816713223, −6.29959618861199613365053970508, −6.12720156536477034196657425095, −5.71727131310871267292831947560, −5.59783094015890141124388957858, −5.02707574767432755245329020092, −4.90579711471960374871692668929, −4.60561826291652367736454190419, −4.06002410212405052187329449350, −3.41927647368554169745515583158, −3.38069576685943282483834129403, −2.65596300850864843339697414290, −2.62429489880858077805972665759, −1.81667089439040978754705944090, −1.45433628058488245717694105534, −0.62305154198281748890644989144, −0.47817342625567856412517645817,
0.47817342625567856412517645817, 0.62305154198281748890644989144, 1.45433628058488245717694105534, 1.81667089439040978754705944090, 2.62429489880858077805972665759, 2.65596300850864843339697414290, 3.38069576685943282483834129403, 3.41927647368554169745515583158, 4.06002410212405052187329449350, 4.60561826291652367736454190419, 4.90579711471960374871692668929, 5.02707574767432755245329020092, 5.59783094015890141124388957858, 5.71727131310871267292831947560, 6.12720156536477034196657425095, 6.29959618861199613365053970508, 6.88976013789666891737816713223, 7.10949935423580399176141247337, 7.52581392362316006552579383772, 7.52945608121675176218474087850
