| L(s) = 1 | + 2-s + 4-s + 5·7-s + 3·8-s − 11-s + 2·13-s + 5·14-s + 16-s − 17-s − 2·19-s − 22-s + 9·23-s + 2·26-s + 5·28-s + 6·29-s + 12·31-s − 32-s − 34-s + 9·37-s − 2·38-s − 3·41-s + 2·43-s − 44-s + 9·46-s + 14·47-s + 9·49-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.88·7-s + 1.06·8-s − 0.301·11-s + 0.554·13-s + 1.33·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.213·22-s + 1.87·23-s + 0.392·26-s + 0.944·28-s + 1.11·29-s + 2.15·31-s − 0.176·32-s − 0.171·34-s + 1.47·37-s − 0.324·38-s − 0.468·41-s + 0.304·43-s − 0.150·44-s + 1.32·46-s + 2.04·47-s + 9/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.496114655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.496114655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 25 T + 294 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 122 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 160 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T - 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−8.921704959136110411258854096121, −8.552060171066060379438389538972, −7.921175777566775559343205342529, −7.81706158639658365913574671241, −7.70413487995788049060092879866, −6.84069148812370687305857169408, −6.78902447456018428628599430273, −6.34293462089370845585102448167, −5.64077965844411046951724034533, −5.54929761566087106478187426473, −4.86259423558116738394237999120, −4.59977486970564661829563120963, −4.30682001607002990553827055923, −4.26371281707278115126355141378, −3.22523769410666306454220606200, −2.77107166179636076775149333695, −2.53024240647249485217131783184, −1.72491052929441478977119378052, −1.34143973199520590204360854964, −0.849040071871901970998004812377,
0.849040071871901970998004812377, 1.34143973199520590204360854964, 1.72491052929441478977119378052, 2.53024240647249485217131783184, 2.77107166179636076775149333695, 3.22523769410666306454220606200, 4.26371281707278115126355141378, 4.30682001607002990553827055923, 4.59977486970564661829563120963, 4.86259423558116738394237999120, 5.54929761566087106478187426473, 5.64077965844411046951724034533, 6.34293462089370845585102448167, 6.78902447456018428628599430273, 6.84069148812370687305857169408, 7.70413487995788049060092879866, 7.81706158639658365913574671241, 7.921175777566775559343205342529, 8.552060171066060379438389538972, 8.921704959136110411258854096121
