| L(s) = 1 | + 3-s + 5-s − 3.41·7-s + 9-s + 2.82·11-s − 2.58·13-s + 15-s − 2.82·17-s + 2.82·19-s − 3.41·21-s − 7.65·23-s + 25-s + 27-s − 1.75·29-s − 31-s + 2.82·33-s − 3.41·35-s − 7.07·37-s − 2.58·39-s + 0.828·41-s − 1.65·43-s + 45-s − 0.343·47-s + 4.65·49-s − 2.82·51-s − 10.8·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.29·7-s + 0.333·9-s + 0.852·11-s − 0.717·13-s + 0.258·15-s − 0.685·17-s + 0.648·19-s − 0.745·21-s − 1.59·23-s + 0.200·25-s + 0.192·27-s − 0.326·29-s − 0.179·31-s + 0.492·33-s − 0.577·35-s − 1.16·37-s − 0.414·39-s + 0.129·41-s − 0.252·43-s + 0.149·45-s − 0.0500·47-s + 0.665·49-s − 0.396·51-s − 1.48·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.58T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 + 2.58T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 - 3.17T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−8.217008015075095843949096022951, −7.33059833703676904021350961973, −6.65319935861018940070959225502, −6.11031708806504296741915259605, −5.14710588458295523999414498658, −4.09326974930617299339188668583, −3.43244087910861355802394332052, −2.54626603073738640722369965510, −1.62922937293437050184542062812, 0,
1.62922937293437050184542062812, 2.54626603073738640722369965510, 3.43244087910861355802394332052, 4.09326974930617299339188668583, 5.14710588458295523999414498658, 6.11031708806504296741915259605, 6.65319935861018940070959225502, 7.33059833703676904021350961973, 8.217008015075095843949096022951
