L(s) = 1 | + 2.20·2-s + 2.83·3-s + 2.85·4-s + 6.25·6-s − 7-s + 1.87·8-s + 5.05·9-s + 4.31·11-s + 8.09·12-s + 1.72·13-s − 2.20·14-s − 1.57·16-s − 2.56·17-s + 11.1·18-s + 2.10·19-s − 2.83·21-s + 9.49·22-s − 23-s + 5.31·24-s + 3.79·26-s + 5.83·27-s − 2.85·28-s + 10.3·29-s + 1.07·31-s − 7.21·32-s + 12.2·33-s − 5.65·34-s + ⋯ |
L(s) = 1 | + 1.55·2-s + 1.63·3-s + 1.42·4-s + 2.55·6-s − 0.377·7-s + 0.662·8-s + 1.68·9-s + 1.29·11-s + 2.33·12-s + 0.478·13-s − 0.588·14-s − 0.393·16-s − 0.623·17-s + 2.62·18-s + 0.483·19-s − 0.619·21-s + 2.02·22-s − 0.208·23-s + 1.08·24-s + 0.744·26-s + 1.12·27-s − 0.538·28-s + 1.91·29-s + 0.192·31-s − 1.27·32-s + 2.12·33-s − 0.970·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.676789541\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.676789541\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 3 | \( 1 - 2.83T + 3T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 + 8.34T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.63T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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
−8.667576372317212298304515224552, −7.59755859955969650532367300795, −6.61087118070752158228106938065, −6.49527835880037680663518459970, −5.22348386480757888234150693000, −4.36403930567142482048706586536, −3.74737713695459266939802634193, −3.21699385278388959696363512012, −2.45896998835737329001706443057, −1.47249139713243530709525069004,
1.47249139713243530709525069004, 2.45896998835737329001706443057, 3.21699385278388959696363512012, 3.74737713695459266939802634193, 4.36403930567142482048706586536, 5.22348386480757888234150693000, 6.49527835880037680663518459970, 6.61087118070752158228106938065, 7.59755859955969650532367300795, 8.667576372317212298304515224552