L(s) = 1 | − 4·25-s + 32·41-s + 48·49-s + 16·89-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-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 | − 4/5·25-s + 4.99·41-s + 48/7·49-s + 1.69·89-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4643536782\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4643536782\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
good | 7 | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.74273883534080419416473394884, −3.50148661074256187630286325854, −3.42580489207215935340862246928, −3.24579827925484386244246930345, −3.05556440908115048893311973353, −3.03160975631890653454451251441, −2.91752414163823111618746903446, −2.90889976952843919419602505395, −2.70664033548921639871243064948, −2.47074106379482017242478664295, −2.30385825412507811582722245515, −2.25809177002569299987594503091, −2.19340794257297939592616934722, −2.10781748291682981416453662346, −2.00027061616746222441317302769, −1.87265941010813181869833285611, −1.73653367681491252106923633426, −1.27425409194433893991340328915, −1.18802285820144614763320698537, −1.07094187060640411065050960137, −0.843796560665312667478461461009, −0.810427589559486585421597239344, −0.73166397763057965766537559657, −0.54476751770788930972896517250, −0.04037428088667642666679374778,
0.04037428088667642666679374778, 0.54476751770788930972896517250, 0.73166397763057965766537559657, 0.810427589559486585421597239344, 0.843796560665312667478461461009, 1.07094187060640411065050960137, 1.18802285820144614763320698537, 1.27425409194433893991340328915, 1.73653367681491252106923633426, 1.87265941010813181869833285611, 2.00027061616746222441317302769, 2.10781748291682981416453662346, 2.19340794257297939592616934722, 2.25809177002569299987594503091, 2.30385825412507811582722245515, 2.47074106379482017242478664295, 2.70664033548921639871243064948, 2.90889976952843919419602505395, 2.91752414163823111618746903446, 3.03160975631890653454451251441, 3.05556440908115048893311973353, 3.24579827925484386244246930345, 3.42580489207215935340862246928, 3.50148661074256187630286325854, 3.74273883534080419416473394884
Plot not available for L-functions of degree greater than 10.