| L(s) = 1 | + 1.73·2-s − 0.732·3-s + 0.999·4-s − 3.34·5-s − 1.26·6-s − 1.41·7-s − 1.73·8-s − 2.46·9-s − 5.79·10-s + 3.46·11-s − 0.732·12-s − 1.93·13-s − 2.44·14-s + 2.44·15-s − 5·16-s + 1.55·17-s − 4.26·18-s + 5.27·19-s − 3.34·20-s + 1.03·21-s + 5.99·22-s + 8.48·23-s + 1.26·24-s + 6.19·25-s − 3.34·26-s + 4·27-s − 1.41·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.422·3-s + 0.499·4-s − 1.49·5-s − 0.517·6-s − 0.534·7-s − 0.612·8-s − 0.821·9-s − 1.83·10-s + 1.04·11-s − 0.211·12-s − 0.535·13-s − 0.654·14-s + 0.632·15-s − 1.25·16-s + 0.376·17-s − 1.00·18-s + 1.21·19-s − 0.748·20-s + 0.225·21-s + 1.27·22-s + 1.76·23-s + 0.258·24-s + 1.23·25-s − 0.656·26-s + 0.769·27-s − 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 97 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 0.656T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 - 0.928T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 - 3.86T + 67T^{2} \) |
| 71 | \( 1 + 5.55T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 0.464T + 89T^{2} \) |
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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
−7.26146974408821273464568664722, −6.50020336775635781931459557836, −5.91098088464374697458612363902, −5.11440451914017922548150126563, −4.61054076994546474899396897472, −3.85001936747242609410430394164, −3.19779230779302513750559855975, −2.83666517085031556575351543779, −1.01851268234506735235878889580, 0,
1.01851268234506735235878889580, 2.83666517085031556575351543779, 3.19779230779302513750559855975, 3.85001936747242609410430394164, 4.61054076994546474899396897472, 5.11440451914017922548150126563, 5.91098088464374697458612363902, 6.50020336775635781931459557836, 7.26146974408821273464568664722
