| L(s) = 1 | + 3-s + 2.61·5-s − 7-s + 9-s − 3.43·11-s + 0.821·13-s + 2.61·15-s − 5.79·17-s − 7.43·19-s − 21-s + 23-s + 1.82·25-s + 27-s − 5.79·29-s − 3.07·31-s − 3.43·33-s − 2.61·35-s − 0.922·37-s + 0.821·39-s + 0.566·41-s − 11.6·43-s + 2.61·45-s − 2.35·47-s + 49-s − 5.79·51-s + 0.968·53-s − 8.96·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.16·5-s − 0.377·7-s + 0.333·9-s − 1.03·11-s + 0.227·13-s + 0.674·15-s − 1.40·17-s − 1.70·19-s − 0.218·21-s + 0.208·23-s + 0.364·25-s + 0.192·27-s − 1.07·29-s − 0.552·31-s − 0.597·33-s − 0.441·35-s − 0.151·37-s + 0.131·39-s + 0.0884·41-s − 1.78·43-s + 0.389·45-s − 0.343·47-s + 0.142·49-s − 0.810·51-s + 0.132·53-s − 1.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2.61T + 5T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 - 0.821T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 + 7.43T + 19T^{2} \) |
| 29 | \( 1 + 5.79T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 + 0.922T + 37T^{2} \) |
| 41 | \( 1 - 0.566T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 - 0.968T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 - 1.17T + 71T^{2} \) |
| 73 | \( 1 - 8.14T + 73T^{2} \) |
| 79 | \( 1 + 2.56T + 79T^{2} \) |
| 83 | \( 1 + 6.30T + 83T^{2} \) |
| 89 | \( 1 + 7.89T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.380036939831698117804452780446, −7.31697270335692530933227410575, −6.59708236807918416555809074184, −5.97513467970114208074076757894, −5.13431762919321298078630388736, −4.28998917873607582524536316708, −3.29336515869630848053944034861, −2.26178818640748139080905645578, −1.89396719079870897524247629410, 0,
1.89396719079870897524247629410, 2.26178818640748139080905645578, 3.29336515869630848053944034861, 4.28998917873607582524536316708, 5.13431762919321298078630388736, 5.97513467970114208074076757894, 6.59708236807918416555809074184, 7.31697270335692530933227410575, 8.380036939831698117804452780446
