| L(s) = 1 | + 2.04e3·4-s + 6.50e4·7-s + 3.14e6·16-s + 4.95e6·19-s + 3.92e6·25-s + 1.33e8·28-s − 4.24e8·43-s + 2.60e9·49-s + 3.21e9·61-s + 4.29e9·64-s − 6.28e9·73-s + 1.01e10·76-s + 8.02e9·100-s + 2.04e11·112-s + 1.04e10·121-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3.87·7-s + 3·16-s + 2·19-s + 0.401·25-s + 7.74·28-s − 2.89·43-s + 9.23·49-s + 3.80·61-s + 4·64-s − 3.03·73-s + 4·76-s + 0.802·100-s + 11.6·112-s + 0.403·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(17.82665538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.82665538\) |
| \(L(6)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3920849 T^{2} + p^{20} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 32525 T + p^{10} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10453237673 T^{2} + p^{20} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 575796429727 T^{2} + p^{20} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 56445243104798 T^{2} + p^{20} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 212457925 T + p^{10} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 103360298822855527 T^{2} + p^{20} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1606836977 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3143217625 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26406402192625404398 T^{2} + p^{20} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
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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
−11.40957322139425022427390081472, −10.90846151284875058486130654387, −10.17228340530200702231450979893, −10.12339256158978153531697229684, −8.807564221205635337798593684480, −8.418102812983138944853011142831, −7.82512236232228664035803839175, −7.76704023360938355533814781968, −7.08872496156640271209488929545, −6.77939227454583927348849999662, −5.61372031520857578839657399637, −5.40029208039584292128266956677, −5.01668054140950256043764935658, −4.29590633485013366948207004338, −3.44614934997188413030387722121, −2.72890845270527415660430866207, −2.02751374633334144747937262699, −1.75892009410399723584490284876, −1.17938333892334643789186665525, −0.973379847261971652308705149417,
0.973379847261971652308705149417, 1.17938333892334643789186665525, 1.75892009410399723584490284876, 2.02751374633334144747937262699, 2.72890845270527415660430866207, 3.44614934997188413030387722121, 4.29590633485013366948207004338, 5.01668054140950256043764935658, 5.40029208039584292128266956677, 5.61372031520857578839657399637, 6.77939227454583927348849999662, 7.08872496156640271209488929545, 7.76704023360938355533814781968, 7.82512236232228664035803839175, 8.418102812983138944853011142831, 8.807564221205635337798593684480, 10.12339256158978153531697229684, 10.17228340530200702231450979893, 10.90846151284875058486130654387, 11.40957322139425022427390081472
