| L(s) = 1 | + 3·3-s − 2·4-s + 6·9-s − 6·12-s + 4·16-s − 10·19-s + 25-s + 9·27-s − 12·36-s + 4·43-s + 12·48-s + 5·49-s − 30·57-s − 8·64-s + 8·67-s + 20·73-s + 3·75-s + 20·76-s + 9·81-s + 12·97-s − 2·100-s − 18·108-s + 121-s + 127-s + 12·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 2·9-s − 1.73·12-s + 16-s − 2.29·19-s + 1/5·25-s + 1.73·27-s − 2·36-s + 0.609·43-s + 1.73·48-s + 5/7·49-s − 3.97·57-s − 64-s + 0.977·67-s + 2.34·73-s + 0.346·75-s + 2.29·76-s + 81-s + 1.21·97-s − 1/5·100-s − 1.73·108-s + 1/11·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.933622057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.933622057\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 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
−8.042951703128934102033379744760, −7.59332520147809094571795914021, −7.07800426008146829374641699848, −6.68109144197669636382141169900, −6.12756610354418670105693511355, −5.70870461113209933596767769840, −4.86730114327834066731559580209, −4.73204334359590177178173372683, −4.01929190912749316951852479878, −3.85434630673308060398055054682, −3.37536387453007675861845473308, −2.61118707760511044268681237936, −2.26490424472686207864944297221, −1.63385937005350265595410279026, −0.65113737512512782605188263400,
0.65113737512512782605188263400, 1.63385937005350265595410279026, 2.26490424472686207864944297221, 2.61118707760511044268681237936, 3.37536387453007675861845473308, 3.85434630673308060398055054682, 4.01929190912749316951852479878, 4.73204334359590177178173372683, 4.86730114327834066731559580209, 5.70870461113209933596767769840, 6.12756610354418670105693511355, 6.68109144197669636382141169900, 7.07800426008146829374641699848, 7.59332520147809094571795914021, 8.042951703128934102033379744760
