| L(s) = 1 | − 4-s + 2·5-s + 4·11-s + 16-s − 12·19-s − 2·20-s − 25-s + 18·29-s − 4·31-s + 22·41-s − 4·44-s + 13·49-s + 8·55-s − 8·59-s − 14·61-s − 64-s + 12·71-s + 12·76-s + 24·79-s + 2·80-s + 2·89-s − 24·95-s + 100-s − 4·101-s − 14·109-s − 18·116-s − 10·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s + 1.20·11-s + 1/4·16-s − 2.75·19-s − 0.447·20-s − 1/5·25-s + 3.34·29-s − 0.718·31-s + 3.43·41-s − 0.603·44-s + 13/7·49-s + 1.07·55-s − 1.04·59-s − 1.79·61-s − 1/8·64-s + 1.42·71-s + 1.37·76-s + 2.70·79-s + 0.223·80-s + 0.211·89-s − 2.46·95-s + 1/10·100-s − 0.398·101-s − 1.34·109-s − 1.67·116-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.154909272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154909272\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.52197178555317896387040196744, −10.04800974545498629268388419157, −9.514610391712751301862732595419, −9.081051582027114513545492207488, −8.953718065377669895966000325791, −8.529198657071138783898618119802, −7.77448706034300682021961158555, −7.75830231638378720781233462230, −6.62177574132703477385759901646, −6.55531831013259950064366906937, −6.20225897781143317383225486233, −5.78117883331226330388537252732, −5.02367800095249948598548689840, −4.56947758903200268615128018521, −4.03473183604154958924012162300, −3.88574082540073227006508084181, −2.60642453590614127625623953694, −2.51285276101068256897793327535, −1.57985110452461566651549498578, −0.77682335440868122790515043342,
0.77682335440868122790515043342, 1.57985110452461566651549498578, 2.51285276101068256897793327535, 2.60642453590614127625623953694, 3.88574082540073227006508084181, 4.03473183604154958924012162300, 4.56947758903200268615128018521, 5.02367800095249948598548689840, 5.78117883331226330388537252732, 6.20225897781143317383225486233, 6.55531831013259950064366906937, 6.62177574132703477385759901646, 7.75830231638378720781233462230, 7.77448706034300682021961158555, 8.529198657071138783898618119802, 8.953718065377669895966000325791, 9.081051582027114513545492207488, 9.514610391712751301862732595419, 10.04800974545498629268388419157, 10.52197178555317896387040196744
