| L(s) = 1 | + 5-s − 9-s + 2·13-s − 2·17-s + 2·29-s − 3·37-s − 2·41-s − 45-s + 4·49-s − 3·53-s + 2·61-s + 2·65-s − 2·73-s − 2·85-s + 3·89-s − 2·97-s + 2·101-s + 2·109-s + 3·113-s − 2·117-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·145-s + 149-s + ⋯ |
| L(s) = 1 | + 5-s − 9-s + 2·13-s − 2·17-s + 2·29-s − 3·37-s − 2·41-s − 45-s + 4·49-s − 3·53-s + 2·61-s + 2·65-s − 2·73-s − 2·85-s + 3·89-s − 2·97-s + 2·101-s + 2·109-s + 3·113-s − 2·117-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·145-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.221823280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221823280\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−7.00371766543096091699328867620, −6.57696057727015355967606272153, −6.44551046213581650169088745613, −6.28398985426557245456819981092, −6.04995159932846715639649453724, −5.88923826015952475583707126015, −5.69520076277469174969051435177, −5.46764742139204314842131343284, −5.24443509450568258219331981418, −4.90580606702507158354179508218, −4.70455789732372464393497721490, −4.60621590431531897335432868065, −4.37528734040920828418377688983, −3.88433363542550711663554815012, −3.71058196575893925915097754418, −3.46109081550391221025307731017, −3.38310452449046606068916386727, −3.01162637887682082632787829172, −2.70502670592038799258822071380, −2.28758774429350980591967667312, −2.24673950601836737448916783596, −1.83983109858557752187143826474, −1.60337412170165041262684511696, −1.22470948941782794062876310414, −0.65014870029216572411316128899,
0.65014870029216572411316128899, 1.22470948941782794062876310414, 1.60337412170165041262684511696, 1.83983109858557752187143826474, 2.24673950601836737448916783596, 2.28758774429350980591967667312, 2.70502670592038799258822071380, 3.01162637887682082632787829172, 3.38310452449046606068916386727, 3.46109081550391221025307731017, 3.71058196575893925915097754418, 3.88433363542550711663554815012, 4.37528734040920828418377688983, 4.60621590431531897335432868065, 4.70455789732372464393497721490, 4.90580606702507158354179508218, 5.24443509450568258219331981418, 5.46764742139204314842131343284, 5.69520076277469174969051435177, 5.88923826015952475583707126015, 6.04995159932846715639649453724, 6.28398985426557245456819981092, 6.44551046213581650169088745613, 6.57696057727015355967606272153, 7.00371766543096091699328867620
