| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s − 4·7-s − 2·8-s + 3·9-s + 2·11-s − 2·12-s − 8·14-s − 4·16-s − 4·17-s + 6·18-s − 12·19-s + 8·21-s + 4·22-s − 4·23-s + 4·24-s − 10·27-s − 4·28-s + 12·29-s − 8·31-s − 2·32-s − 4·33-s − 8·34-s + 3·36-s + 32·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s − 1.51·7-s − 0.707·8-s + 9-s + 0.603·11-s − 0.577·12-s − 2.13·14-s − 16-s − 0.970·17-s + 1.41·18-s − 2.75·19-s + 1.74·21-s + 0.852·22-s − 0.834·23-s + 0.816·24-s − 1.92·27-s − 0.755·28-s + 2.22·29-s − 1.43·31-s − 0.353·32-s − 0.696·33-s − 1.37·34-s + 1/2·36-s + 5.26·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.616380583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616380583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 10 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 - 20 T^{2} + 231 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 80 T^{3} - 221 T^{4} - 80 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 80 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 190 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 76 T^{2} + 8 T^{3} + 5503 T^{4} + 8 p T^{5} - 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 2 T + 35 T^{2} - 298 T^{3} - 2756 T^{4} - 298 p T^{5} + 35 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 8 p T^{5} - 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 19 T^{2} + 602 T^{3} + 13708 T^{4} + 602 p T^{5} + 19 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 104 T^{2} + 4575 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.942057358324840163283754998058, −7.87669525043433559340005638348, −7.40384526576942493299076264572, −7.12908128955509097138178391139, −6.73753042409328170542425827272, −6.51180642990837421445938485240, −6.50661387261909198925683371210, −6.18255658830225352018272243460, −6.06282639687909995976393198004, −5.93859080442780097752189167097, −5.45079880575058505738232033927, −5.40260925178846966322644000131, −4.73249618368452611864812295828, −4.70347292705058828804075686382, −4.29490079782177089485085631554, −4.13327770580837423371503938398, −4.03039882103822475083830124876, −3.81502647006999228312765959573, −3.17759902250341638370801705926, −3.03791970702449874563586400003, −2.33978666850735247412881546720, −2.29049521467912837159803230977, −1.98725334055698346864776450894, −0.842468874971710756379977911906, −0.50842669699888039317717842944,
0.50842669699888039317717842944, 0.842468874971710756379977911906, 1.98725334055698346864776450894, 2.29049521467912837159803230977, 2.33978666850735247412881546720, 3.03791970702449874563586400003, 3.17759902250341638370801705926, 3.81502647006999228312765959573, 4.03039882103822475083830124876, 4.13327770580837423371503938398, 4.29490079782177089485085631554, 4.70347292705058828804075686382, 4.73249618368452611864812295828, 5.40260925178846966322644000131, 5.45079880575058505738232033927, 5.93859080442780097752189167097, 6.06282639687909995976393198004, 6.18255658830225352018272243460, 6.50661387261909198925683371210, 6.51180642990837421445938485240, 6.73753042409328170542425827272, 7.12908128955509097138178391139, 7.40384526576942493299076264572, 7.87669525043433559340005638348, 7.942057358324840163283754998058
