| L(s) = 1 | − 8·4-s + 40·16-s + 28·49-s − 160·64-s + 10·81-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s − 224·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4·4-s + 10·16-s + 4·49-s − 20·64-s + 10/9·81-s − 8·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 16·196-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3430750726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3430750726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} )^{4} \) |
| 3 | \( 1 - 10 T^{4} + p^{4} T^{8} \) |
| 7 | \( ( 1 - p T^{2} )^{4} \) |
| good | 5 | \( ( 1 + 22 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 + 310 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 - 650 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 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 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 1130 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 7370 T^{4} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 13130 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−5.63570856612103873034045206863, −5.55069630313077150408070311816, −5.45358464703962491850179723483, −5.41948169760209339774001169767, −5.38420495821886226800928382031, −5.03857142588052947880103222464, −4.82497503698943646906338391506, −4.71011164419727524042511196043, −4.69480786319982795740479484200, −4.31870903147471137412630187115, −4.25294849207653958435038262608, −4.05588669246356563725010332985, −3.89135341580625531500095494698, −3.72715325844416900736938918670, −3.72701974364572639745311570147, −3.34275970871489480947168112588, −3.30789931101140729284184083669, −2.94718055445782401687889657671, −2.52148227824101171287759760405, −2.45235352092706127272020359547, −2.33356906938141667109848102754, −1.42638230889915777891905303173, −1.37254558373175719219593004940, −1.04038786853832952868559517285, −0.40116088426021945553881671768,
0.40116088426021945553881671768, 1.04038786853832952868559517285, 1.37254558373175719219593004940, 1.42638230889915777891905303173, 2.33356906938141667109848102754, 2.45235352092706127272020359547, 2.52148227824101171287759760405, 2.94718055445782401687889657671, 3.30789931101140729284184083669, 3.34275970871489480947168112588, 3.72701974364572639745311570147, 3.72715325844416900736938918670, 3.89135341580625531500095494698, 4.05588669246356563725010332985, 4.25294849207653958435038262608, 4.31870903147471137412630187115, 4.69480786319982795740479484200, 4.71011164419727524042511196043, 4.82497503698943646906338391506, 5.03857142588052947880103222464, 5.38420495821886226800928382031, 5.41948169760209339774001169767, 5.45358464703962491850179723483, 5.55069630313077150408070311816, 5.63570856612103873034045206863
Plot not available for L-functions of degree greater than 10.
