| L(s) = 1 | − 3-s − 5-s + 3.77·7-s + 9-s + 3.60·11-s − 5.22·13-s + 15-s + 1.62·17-s − 3.15·19-s − 3.77·21-s − 6.55·23-s + 25-s − 27-s + 1.87·29-s − 9.40·31-s − 3.60·33-s − 3.77·35-s + 1.90·37-s + 5.22·39-s − 10.8·41-s + 0.674·43-s − 45-s − 1.78·47-s + 7.26·49-s − 1.62·51-s + 6.01·53-s − 3.60·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.42·7-s + 0.333·9-s + 1.08·11-s − 1.44·13-s + 0.258·15-s + 0.393·17-s − 0.723·19-s − 0.824·21-s − 1.36·23-s + 0.200·25-s − 0.192·27-s + 0.348·29-s − 1.68·31-s − 0.627·33-s − 0.638·35-s + 0.312·37-s + 0.836·39-s − 1.68·41-s + 0.102·43-s − 0.149·45-s − 0.259·47-s + 1.03·49-s − 0.227·51-s + 0.825·53-s − 0.485·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 + 5.22T + 13T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 + 6.55T + 23T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 + 9.40T + 31T^{2} \) |
| 37 | \( 1 - 1.90T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 0.674T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 - 6.01T + 53T^{2} \) |
| 59 | \( 1 + 0.839T + 59T^{2} \) |
| 61 | \( 1 + 2.26T + 61T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 - 7.98T + 79T^{2} \) |
| 83 | \( 1 - 4.72T + 83T^{2} \) |
| 89 | \( 1 + 2.96T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
| 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
−8.004167326213336947893674852422, −7.37199709140051831237059960996, −6.70824507554823837697950804265, −5.74638262686621925923195898925, −5.00232762181245347179980683183, −4.38964686724755822723205940514, −3.66971489916558153900641823155, −2.21267088216469193599465764249, −1.45215772366569019935479932037, 0,
1.45215772366569019935479932037, 2.21267088216469193599465764249, 3.66971489916558153900641823155, 4.38964686724755822723205940514, 5.00232762181245347179980683183, 5.74638262686621925923195898925, 6.70824507554823837697950804265, 7.37199709140051831237059960996, 8.004167326213336947893674852422
