L(s) = 1 | − 2·13-s − 2·17-s − 8·37-s − 8·53-s + 2·73-s + 81-s + 10·89-s + 2·97-s + 4·101-s − 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | − 2·13-s − 2·17-s − 8·37-s − 8·53-s + 2·73-s + 81-s + 10·89-s + 2·97-s + 4·101-s − 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2225169021\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2225169021\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 13 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 17 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 37 | \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 53 | \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 67 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 89 | \( ( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.64287339607775174670315998273, −3.55697033631481768419599102173, −3.42625349440374281574296387389, −3.32809014689970200764436480556, −3.29099709857617645181734339917, −3.27606640128429834635803549029, −3.09868494906398745957963937224, −3.00156338490065672218981955566, −2.66886965184114505833188087916, −2.66769380532270550784370097460, −2.46867510874005352708696823966, −2.36816929672583738871697376756, −2.07700804261814804163447586923, −2.07116336530858488929950858830, −2.00469962958422626197963829526, −1.96819925723893023740312552428, −1.93138120635499229201319970850, −1.69603992315337827643740621500, −1.48825040359220765606321554022, −1.43870518205962038865426797308, −1.40918208346695072936083268714, −0.842515910366969655842917993558, −0.66706664084150803662014498185, −0.57895258705990064752217201139, −0.14373568290828362381687220266,
0.14373568290828362381687220266, 0.57895258705990064752217201139, 0.66706664084150803662014498185, 0.842515910366969655842917993558, 1.40918208346695072936083268714, 1.43870518205962038865426797308, 1.48825040359220765606321554022, 1.69603992315337827643740621500, 1.93138120635499229201319970850, 1.96819925723893023740312552428, 2.00469962958422626197963829526, 2.07116336530858488929950858830, 2.07700804261814804163447586923, 2.36816929672583738871697376756, 2.46867510874005352708696823966, 2.66769380532270550784370097460, 2.66886965184114505833188087916, 3.00156338490065672218981955566, 3.09868494906398745957963937224, 3.27606640128429834635803549029, 3.29099709857617645181734339917, 3.32809014689970200764436480556, 3.42625349440374281574296387389, 3.55697033631481768419599102173, 3.64287339607775174670315998273
Plot not available for L-functions of degree greater than 10.