| L(s) = 1 | − 2·3-s − 4-s − 6·11-s + 2·12-s − 3·16-s + 12·17-s + 4·19-s + 6·23-s − 7·25-s + 2·27-s − 6·29-s − 2·31-s + 12·33-s + 14·37-s + 10·43-s + 6·44-s + 12·47-s + 6·48-s − 24·51-s − 6·53-s − 8·57-s + 18·59-s + 20·61-s + 7·64-s + 2·67-s − 12·68-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 1.80·11-s + 0.577·12-s − 3/4·16-s + 2.91·17-s + 0.917·19-s + 1.25·23-s − 7/5·25-s + 0.384·27-s − 1.11·29-s − 0.359·31-s + 2.08·33-s + 2.30·37-s + 1.52·43-s + 0.904·44-s + 1.75·47-s + 0.866·48-s − 3.36·51-s − 0.824·53-s − 1.05·57-s + 2.34·59-s + 2.56·61-s + 7/8·64-s + 0.244·67-s − 1.45·68-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683063315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683063315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 195 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 123 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.74278921476140862347399772201, −7.66355819136458177168709846198, −7.29267059921289013741027929179, −7.25434189455838319139100948948, −6.38474869280347825547736784193, −6.09356677870107854196906219099, −5.78657018617395393221871111161, −5.45598887328111590311392728950, −5.22183742436540988300595869646, −5.19549790539972441915303791422, −4.63478160966646967018828424512, −4.05246919670495923940146453428, −3.66260283536706829563095684577, −3.45551201068471328653941765622, −2.82987671572009940831867860886, −2.34949158270939851314979319349, −2.23492256700591907414137560934, −1.18326770410323965203014203930, −0.68549613140077777341186965782, −0.56409947913096698444824593142,
0.56409947913096698444824593142, 0.68549613140077777341186965782, 1.18326770410323965203014203930, 2.23492256700591907414137560934, 2.34949158270939851314979319349, 2.82987671572009940831867860886, 3.45551201068471328653941765622, 3.66260283536706829563095684577, 4.05246919670495923940146453428, 4.63478160966646967018828424512, 5.19549790539972441915303791422, 5.22183742436540988300595869646, 5.45598887328111590311392728950, 5.78657018617395393221871111161, 6.09356677870107854196906219099, 6.38474869280347825547736784193, 7.25434189455838319139100948948, 7.29267059921289013741027929179, 7.66355819136458177168709846198, 7.74278921476140862347399772201
