| L(s) = 1 | − 16·4-s − 27·9-s + 192·16-s − 112·19-s − 250·25-s + 432·36-s − 220·37-s − 286·49-s − 2.04e3·64-s + 880·67-s − 2.38e3·73-s + 1.79e3·76-s + 729·81-s + 4.00e3·100-s − 3.64e3·103-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s − 5.18e3·144-s + 3.52e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s + 3·16-s − 1.35·19-s − 2·25-s + 2·36-s − 0.977·37-s − 0.833·49-s − 4·64-s + 1.60·67-s − 3.81·73-s + 2.70·76-s + 81-s + 4·100-s − 3.48·103-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3·144-s + 1.95·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04924931875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04924931875\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| 67 | $C_2$ | \( 1 - 880 T + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 20 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )( 1 + 520 T + p^{3} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )( 1 + 182 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1190 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )( 1 + 884 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \) |
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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
−12.57965691451120960661235171515, −11.70197924022879915118676770121, −11.44301739480388808777812115562, −10.44031076525498834467129316778, −10.38549000797420859598069883556, −9.601211249118831312829437356598, −9.281383711977706976661445029648, −8.756596408526256011034653092661, −8.209362677422026480680892802523, −8.113291644496579265846204380921, −7.28188507526970605936651403810, −6.33780361926974888504677185224, −5.81170062061517169817135730251, −5.34860825482106459201274946761, −4.72165031413561355605528058130, −4.02595647914943665886086498940, −3.67670025282945162745938213002, −2.70075135303811268671558316027, −1.51240198732345375275600992910, −0.10053276823414201082055138696,
0.10053276823414201082055138696, 1.51240198732345375275600992910, 2.70075135303811268671558316027, 3.67670025282945162745938213002, 4.02595647914943665886086498940, 4.72165031413561355605528058130, 5.34860825482106459201274946761, 5.81170062061517169817135730251, 6.33780361926974888504677185224, 7.28188507526970605936651403810, 8.113291644496579265846204380921, 8.209362677422026480680892802523, 8.756596408526256011034653092661, 9.281383711977706976661445029648, 9.601211249118831312829437356598, 10.38549000797420859598069883556, 10.44031076525498834467129316778, 11.44301739480388808777812115562, 11.70197924022879915118676770121, 12.57965691451120960661235171515
