| L(s) = 1 | − 0.848·3-s + 5-s − 3.71·7-s − 2.27·9-s − 1.68·11-s − 0.240·13-s − 0.848·15-s + 6.93·17-s + 0.373·19-s + 3.15·21-s + 4.19·23-s + 25-s + 4.48·27-s − 7.90·29-s + 5.14·31-s + 1.42·33-s − 3.71·35-s − 4.32·37-s + 0.204·39-s − 8.86·41-s + 2.98·43-s − 2.27·45-s + 5.46·47-s + 6.79·49-s − 5.88·51-s + 0.449·53-s − 1.68·55-s + ⋯ |
| L(s) = 1 | − 0.489·3-s + 0.447·5-s − 1.40·7-s − 0.759·9-s − 0.506·11-s − 0.0666·13-s − 0.219·15-s + 1.68·17-s + 0.0856·19-s + 0.687·21-s + 0.875·23-s + 0.200·25-s + 0.862·27-s − 1.46·29-s + 0.923·31-s + 0.248·33-s − 0.627·35-s − 0.711·37-s + 0.0326·39-s − 1.38·41-s + 0.455·43-s − 0.339·45-s + 0.796·47-s + 0.970·49-s − 0.823·51-s + 0.0617·53-s − 0.226·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 0.848T + 3T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 0.240T + 13T^{2} \) |
| 17 | \( 1 - 6.93T + 17T^{2} \) |
| 19 | \( 1 - 0.373T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 + 7.90T + 29T^{2} \) |
| 31 | \( 1 - 5.14T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 + 8.86T + 41T^{2} \) |
| 43 | \( 1 - 2.98T + 43T^{2} \) |
| 47 | \( 1 - 5.46T + 47T^{2} \) |
| 53 | \( 1 - 0.449T + 53T^{2} \) |
| 59 | \( 1 - 1.77T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 + 3.00T + 71T^{2} \) |
| 73 | \( 1 + 4.50T + 73T^{2} \) |
| 79 | \( 1 - 2.72T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 - 7.60T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33691858747253783318521376992, −6.72990600902737184671921436188, −5.97538417348438222641602771114, −5.53024766059593834639497707470, −4.96776632102636886293497747019, −3.63756112696088164588425765582, −3.16305856497871572459222221531, −2.39590943354261726570996078956, −1.05001294357639571874580016142, 0,
1.05001294357639571874580016142, 2.39590943354261726570996078956, 3.16305856497871572459222221531, 3.63756112696088164588425765582, 4.96776632102636886293497747019, 5.53024766059593834639497707470, 5.97538417348438222641602771114, 6.72990600902737184671921436188, 7.33691858747253783318521376992
