| L(s) = 1 | − 4·3-s + 4·4-s + 6·9-s − 16·12-s + 12·16-s + 12·17-s − 6·23-s + 25-s + 4·27-s − 18·29-s + 24·36-s + 2·43-s − 48·48-s − 49-s − 48·51-s − 18·53-s − 20·61-s + 32·64-s + 48·68-s + 24·69-s − 4·75-s − 2·79-s − 37·81-s + 72·87-s − 24·92-s + 4·100-s + 8·103-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·4-s + 2·9-s − 4.61·12-s + 3·16-s + 2.91·17-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 3.34·29-s + 4·36-s + 0.304·43-s − 6.92·48-s − 1/7·49-s − 6.72·51-s − 2.47·53-s − 2.56·61-s + 4·64-s + 5.82·68-s + 2.88·69-s − 0.461·75-s − 0.225·79-s − 4.11·81-s + 7.71·87-s − 2.50·92-s + 2/5·100-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294572703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294572703\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−10.24277281295338639514036251995, −9.685071767104213573392884697364, −9.575889413916126804594831808033, −8.677341569576823091609854290186, −7.943809659807535129151592911537, −7.65590970416138626610152838860, −7.54592372411120679323584788208, −6.97263834682839771865531714548, −6.40517329576887599962897624991, −6.05029007211674608258170698378, −5.89745373132924580817223792627, −5.40272482687157513786812509579, −5.38938690379834626450161030837, −4.55724141606123843471714834302, −3.71299663737923261566887746202, −3.23901684986306267091865053510, −2.87052458845859655850716669971, −1.65555022703813176395529685143, −1.63238860924544709848480149261, −0.56248222403482296849022218812,
0.56248222403482296849022218812, 1.63238860924544709848480149261, 1.65555022703813176395529685143, 2.87052458845859655850716669971, 3.23901684986306267091865053510, 3.71299663737923261566887746202, 4.55724141606123843471714834302, 5.38938690379834626450161030837, 5.40272482687157513786812509579, 5.89745373132924580817223792627, 6.05029007211674608258170698378, 6.40517329576887599962897624991, 6.97263834682839771865531714548, 7.54592372411120679323584788208, 7.65590970416138626610152838860, 7.943809659807535129151592911537, 8.677341569576823091609854290186, 9.575889413916126804594831808033, 9.685071767104213573392884697364, 10.24277281295338639514036251995
