| L(s) = 1 | + 3·3-s − 0.128·5-s + 9·9-s − 54.0·11-s + 50.2·13-s − 0.385·15-s + 131.·17-s + 91.5·19-s − 179.·23-s − 124.·25-s + 27·27-s − 69.8·29-s + 326.·31-s − 162.·33-s + 301.·37-s + 150.·39-s − 296.·41-s + 144.·43-s − 1.15·45-s − 360.·47-s + 394.·51-s + 1.83·53-s + 6.94·55-s + 274.·57-s − 53.2·59-s + 108.·61-s − 6.45·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0115·5-s + 0.333·9-s − 1.48·11-s + 1.07·13-s − 0.00664·15-s + 1.87·17-s + 1.10·19-s − 1.62·23-s − 0.999·25-s + 0.192·27-s − 0.447·29-s + 1.89·31-s − 0.854·33-s + 1.34·37-s + 0.618·39-s − 1.12·41-s + 0.511·43-s − 0.00383·45-s − 1.11·47-s + 1.08·51-s + 0.00475·53-s + 0.0170·55-s + 0.637·57-s − 0.117·59-s + 0.226·61-s − 0.0123·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.846644877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.846644877\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.128T + 125T^{2} \) |
| 11 | \( 1 + 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 326.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 296.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 360.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.83T + 1.48e5T^{2} \) |
| 59 | \( 1 + 53.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 842.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 241.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 559.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 986.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 740.T + 9.12e5T^{2} \) |
| 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
−8.241202196457079442611570594576, −8.047628852300257688429149033770, −7.42326734021609721974967477058, −6.10232230444694308002228141400, −5.62744685557393856436974677125, −4.60678686698642730106025423198, −3.54384156520016549761878239828, −2.95854391324749404882833279229, −1.83508609241373917668012029965, −0.73953391487158070163236334724,
0.73953391487158070163236334724, 1.83508609241373917668012029965, 2.95854391324749404882833279229, 3.54384156520016549761878239828, 4.60678686698642730106025423198, 5.62744685557393856436974677125, 6.10232230444694308002228141400, 7.42326734021609721974967477058, 8.047628852300257688429149033770, 8.241202196457079442611570594576
