| L(s) = 1 | + 4·9-s − 32·25-s − 8·41-s + 104·73-s − 26·81-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 76·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 128·225-s + 227-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 6.39·25-s − 1.24·41-s + 12.1·73-s − 2.88·81-s + 7.27·121-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 + 5.84·169-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 − 8.53·225-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.274368185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.274368185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 51 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 47 | \( ( 1 - 69 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 13 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.88555339067288743096506036322, −3.80867141027606773847003988489, −3.80321846009955105052128994482, −3.77946257362785574313186395133, −3.71624822847505334300608515569, −3.44566049910370577642468499238, −3.24519692117014510431754410941, −3.24138292521805466659936148813, −3.05862671352208141387257725909, −2.84751592974287806992818794876, −2.69713200151202728543797055281, −2.57857594229707287742293372163, −2.28226036717206368259492394773, −2.08989977841661076993352253733, −2.06261749815826485755825260002, −2.01638879718668537688638687674, −1.80724782660616986673268046702, −1.74323939952502666254343721955, −1.71491505850444883637576547118, −1.53077192848458262126951548039, −0.987430062212206418438536403094, −0.77170526424234955484223610520, −0.72157673415897621769450685303, −0.52665422816923060111751774495, −0.25021838095409725987334429615,
0.25021838095409725987334429615, 0.52665422816923060111751774495, 0.72157673415897621769450685303, 0.77170526424234955484223610520, 0.987430062212206418438536403094, 1.53077192848458262126951548039, 1.71491505850444883637576547118, 1.74323939952502666254343721955, 1.80724782660616986673268046702, 2.01638879718668537688638687674, 2.06261749815826485755825260002, 2.08989977841661076993352253733, 2.28226036717206368259492394773, 2.57857594229707287742293372163, 2.69713200151202728543797055281, 2.84751592974287806992818794876, 3.05862671352208141387257725909, 3.24138292521805466659936148813, 3.24519692117014510431754410941, 3.44566049910370577642468499238, 3.71624822847505334300608515569, 3.77946257362785574313186395133, 3.80321846009955105052128994482, 3.80867141027606773847003988489, 3.88555339067288743096506036322
Plot not available for L-functions of degree greater than 10.
