| L(s) = 1 | + 68·11-s − 328·19-s − 292·29-s + 200·31-s − 576·41-s + 362·49-s − 1.28e3·59-s + 1.20e3·61-s + 1.30e3·71-s − 776·79-s + 1.64e3·89-s − 1.59e3·101-s + 396·109-s + 806·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.25e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.86·11-s − 3.96·19-s − 1.86·29-s + 1.15·31-s − 2.19·41-s + 1.05·49-s − 2.83·59-s + 2.52·61-s + 2.17·71-s − 1.10·79-s + 1.95·89-s − 1.57·101-s + 0.347·109-s + 0.605·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.93·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09941495889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09941495889\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 362 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22030 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 146 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6278 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 288 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 144614 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 207390 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 281878 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 642 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 602 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 411430 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 652 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 349810 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 388 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 946438 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 820 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1238590 T^{2} + p^{6} T^{4} \) |
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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
−9.168876540196448358245094479890, −8.669003200052254935327035754423, −8.504583306019125323992786528694, −7.991687234864197156832767677674, −7.62460883813928955829211532162, −6.78534477601093108375514833334, −6.78107458805227617259886829369, −6.31936753128841503230634265005, −6.20800618494999811490327258087, −5.48090354492161433327935771825, −5.01982741804306520870234579726, −4.30374673589243183873231245239, −4.28684435985304651758825766701, −3.66506049336864269401826874791, −3.48635298076517765561621587986, −2.31219757503544581686062537829, −2.29202952162701447076323758834, −1.60770701664398782225133602978, −1.10468727726316484912203662038, −0.06784971520862014483734632499,
0.06784971520862014483734632499, 1.10468727726316484912203662038, 1.60770701664398782225133602978, 2.29202952162701447076323758834, 2.31219757503544581686062537829, 3.48635298076517765561621587986, 3.66506049336864269401826874791, 4.28684435985304651758825766701, 4.30374673589243183873231245239, 5.01982741804306520870234579726, 5.48090354492161433327935771825, 6.20800618494999811490327258087, 6.31936753128841503230634265005, 6.78107458805227617259886829369, 6.78534477601093108375514833334, 7.62460883813928955829211532162, 7.991687234864197156832767677674, 8.504583306019125323992786528694, 8.669003200052254935327035754423, 9.168876540196448358245094479890
