| L(s) = 1 | − 2·16-s − 4·49-s − 81-s − 8·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 2·16-s − 4·49-s − 81-s − 8·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5137390383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5137390383\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | | \( 1 \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71032443837427956329442847953, −6.69436151736711633518903983827, −6.26222535343257359988756493995, −5.91704177012419885978283356177, −5.90567529884247538508891719902, −5.68376103870545192860691440655, −5.19891902844400130386375685940, −5.12524807159713461958350129817, −5.08215947008945072946602121854, −4.73682795611349426892386075174, −4.53279872115235031684175581565, −4.20865028647409363998692113396, −4.15364416745678358955646277686, −3.97060540279298605551049685533, −3.54128174111645860228937893763, −3.34302047631195795110182512882, −3.12826316079474454733365428903, −2.72157005976622050768543625316, −2.59021122765571353649079088826, −2.48864770213640067956031190971, −1.97754050937281310951466859758, −1.67862085979285519302565276236, −1.47067445861492531573189272021, −1.22802583446277363449727970081, −0.33228161875003712152339817706,
0.33228161875003712152339817706, 1.22802583446277363449727970081, 1.47067445861492531573189272021, 1.67862085979285519302565276236, 1.97754050937281310951466859758, 2.48864770213640067956031190971, 2.59021122765571353649079088826, 2.72157005976622050768543625316, 3.12826316079474454733365428903, 3.34302047631195795110182512882, 3.54128174111645860228937893763, 3.97060540279298605551049685533, 4.15364416745678358955646277686, 4.20865028647409363998692113396, 4.53279872115235031684175581565, 4.73682795611349426892386075174, 5.08215947008945072946602121854, 5.12524807159713461958350129817, 5.19891902844400130386375685940, 5.68376103870545192860691440655, 5.90567529884247538508891719902, 5.91704177012419885978283356177, 6.26222535343257359988756493995, 6.69436151736711633518903983827, 6.71032443837427956329442847953
