| L(s) = 1 | − 3-s + 0.585·5-s − 3.41·7-s + 9-s − 2·11-s + 2.82·13-s − 0.585·15-s + 3.65·17-s + 5.65·19-s + 3.41·21-s − 1.17·23-s − 4.65·25-s − 27-s + 0.585·29-s − 4.58·31-s + 2·33-s − 2·35-s − 9.65·37-s − 2.82·39-s − 11.6·41-s − 1.65·43-s + 0.585·45-s − 12.4·47-s + 4.65·49-s − 3.65·51-s + 11.8·53-s − 1.17·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.261·5-s − 1.29·7-s + 0.333·9-s − 0.603·11-s + 0.784·13-s − 0.151·15-s + 0.886·17-s + 1.29·19-s + 0.745·21-s − 0.244·23-s − 0.931·25-s − 0.192·27-s + 0.108·29-s − 0.823·31-s + 0.348·33-s − 0.338·35-s − 1.58·37-s − 0.452·39-s − 1.82·41-s − 0.252·43-s + 0.0873·45-s − 1.82·47-s + 0.665·49-s − 0.512·51-s + 1.63·53-s − 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 9.65T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 9.17T + 71T^{2} \) |
| 73 | \( 1 + 1.65T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.227805000838494247216220735257, −8.249573302753600809777390498565, −7.28662420213233525916928170900, −6.57293713105416332314713311281, −5.69593168188394116934774112579, −5.21095925852256138048589516863, −3.71787849065387894947905118136, −3.12536603370223893837381633932, −1.56426393760888427293250305267, 0,
1.56426393760888427293250305267, 3.12536603370223893837381633932, 3.71787849065387894947905118136, 5.21095925852256138048589516863, 5.69593168188394116934774112579, 6.57293713105416332314713311281, 7.28662420213233525916928170900, 8.249573302753600809777390498565, 9.227805000838494247216220735257
