| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 4·14-s + 16-s + 2·25-s + 4·28-s + 4·29-s − 32-s − 8·41-s + 8·43-s + 2·49-s − 2·50-s − 4·56-s − 4·58-s − 4·59-s + 64-s − 12·71-s + 12·73-s + 8·82-s − 8·86-s + 20·89-s − 2·98-s + 2·100-s + 4·107-s + 4·112-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1.06·14-s + 1/4·16-s + 2/5·25-s + 0.755·28-s + 0.742·29-s − 0.176·32-s − 1.24·41-s + 1.21·43-s + 2/7·49-s − 0.282·50-s − 0.534·56-s − 0.525·58-s − 0.520·59-s + 1/8·64-s − 1.42·71-s + 1.40·73-s + 0.883·82-s − 0.862·86-s + 2.11·89-s − 0.202·98-s + 1/5·100-s + 0.386·107-s + 0.377·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657038713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657038713\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.422820232220852815671553534810, −8.217663777026007329140330631404, −7.70199990091947954567948088263, −7.39388351910402354458306519270, −6.78350396560819892745171391544, −6.37946188618351970821569126314, −5.76419845422234777580939390754, −5.26880397922174816557603358477, −4.68626329432089487067401150982, −4.43427680066925394093217561381, −3.54218152983186298415138093143, −2.97291040152222612448694683527, −2.15470271326811965467737354033, −1.65861118208639812124511823643, −0.810877027849803187742138121374,
0.810877027849803187742138121374, 1.65861118208639812124511823643, 2.15470271326811965467737354033, 2.97291040152222612448694683527, 3.54218152983186298415138093143, 4.43427680066925394093217561381, 4.68626329432089487067401150982, 5.26880397922174816557603358477, 5.76419845422234777580939390754, 6.37946188618351970821569126314, 6.78350396560819892745171391544, 7.39388351910402354458306519270, 7.70199990091947954567948088263, 8.217663777026007329140330631404, 8.422820232220852815671553534810
