| L(s) = 1 | + 2·9-s + 20·25-s − 36·29-s − 28·49-s + 19·81-s + 44·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 40·225-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 4·25-s − 6.68·29-s − 4·49-s + 19/9·81-s + 4.21·109-s + 2.36·121-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 − 2.92·169-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 + 8/3·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2703813253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2703813253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 19 T^{2} + 192 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 29 T^{2} + 552 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 31 T^{2} - 1248 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2}( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 - 149 T^{2} + 12792 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.82292084863837207629945692964, −3.69047640252624943316200117892, −3.54342535811374369297012736474, −3.45623074171013704168866921721, −3.23518706687415579094348172684, −3.14316752830230600497394577013, −3.09061268317235211881963558598, −3.02993955418780047744115626317, −2.94314048409283873834380632228, −2.66415773827890002852940406719, −2.42441560630674281224211086841, −2.30055851495701987458672385697, −2.22511007179534896046565364104, −2.20360587676510659218615833388, −1.93497822160919932889858279781, −1.67554621330053127735000531562, −1.65508066099385547718730629210, −1.62139850097163927499754755689, −1.55827402865389829302806693561, −1.23484645323034467890196917071, −0.886361556371150594387757813358, −0.832008694225358354106338651092, −0.77748425798194555877518200626, −0.28784622818536308819919780910, −0.05859793494028838255593766708,
0.05859793494028838255593766708, 0.28784622818536308819919780910, 0.77748425798194555877518200626, 0.832008694225358354106338651092, 0.886361556371150594387757813358, 1.23484645323034467890196917071, 1.55827402865389829302806693561, 1.62139850097163927499754755689, 1.65508066099385547718730629210, 1.67554621330053127735000531562, 1.93497822160919932889858279781, 2.20360587676510659218615833388, 2.22511007179534896046565364104, 2.30055851495701987458672385697, 2.42441560630674281224211086841, 2.66415773827890002852940406719, 2.94314048409283873834380632228, 3.02993955418780047744115626317, 3.09061268317235211881963558598, 3.14316752830230600497394577013, 3.23518706687415579094348172684, 3.45623074171013704168866921721, 3.54342535811374369297012736474, 3.69047640252624943316200117892, 3.82292084863837207629945692964
Plot not available for L-functions of degree greater than 10.
