| L(s) = 1 | − 1.24·2-s − 3-s − 0.445·4-s + 3.06·5-s + 1.24·6-s + 7-s + 3.04·8-s + 9-s − 3.82·10-s − 3.50·11-s + 0.445·12-s − 1.24·14-s − 3.06·15-s − 2.91·16-s + 7.13·17-s − 1.24·18-s − 3.36·19-s − 1.36·20-s − 21-s + 4.37·22-s − 1.88·23-s − 3.04·24-s + 4.42·25-s − 27-s − 0.445·28-s − 4.40·29-s + 3.82·30-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 0.577·3-s − 0.222·4-s + 1.37·5-s + 0.509·6-s + 0.377·7-s + 1.07·8-s + 0.333·9-s − 1.21·10-s − 1.05·11-s + 0.128·12-s − 0.333·14-s − 0.792·15-s − 0.727·16-s + 1.73·17-s − 0.293·18-s − 0.772·19-s − 0.305·20-s − 0.218·21-s + 0.931·22-s − 0.392·23-s − 0.622·24-s + 0.884·25-s − 0.192·27-s − 0.0841·28-s − 0.817·29-s + 0.698·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 17 | \( 1 - 7.13T + 17T^{2} \) |
| 19 | \( 1 + 3.36T + 19T^{2} \) |
| 23 | \( 1 + 1.88T + 23T^{2} \) |
| 29 | \( 1 + 4.40T + 29T^{2} \) |
| 31 | \( 1 + 2.60T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 6.41T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 1.93T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 4.91T + 73T^{2} \) |
| 79 | \( 1 + 3.41T + 79T^{2} \) |
| 83 | \( 1 - 7.10T + 83T^{2} \) |
| 89 | \( 1 + 0.730T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
| 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
−8.131584827943877339605129656995, −7.72700173026312142335144407821, −6.75143374363011940385955814917, −5.79888294699642755266999003768, −5.32195815480151314916484893941, −4.65387510329205918720138819870, −3.37588065613025603029244684813, −2.03183618760848178786452835754, −1.39976852590423715697623086716, 0,
1.39976852590423715697623086716, 2.03183618760848178786452835754, 3.37588065613025603029244684813, 4.65387510329205918720138819870, 5.32195815480151314916484893941, 5.79888294699642755266999003768, 6.75143374363011940385955814917, 7.72700173026312142335144407821, 8.131584827943877339605129656995
