| L(s) = 1 | + 0.0903·2-s + 1.28·3-s − 1.99·4-s + 0.116·6-s − 7-s − 0.360·8-s − 1.34·9-s − 3.84·11-s − 2.56·12-s − 5.39·13-s − 0.0903·14-s + 3.95·16-s + 5.02·17-s − 0.121·18-s + 19-s − 1.28·21-s − 0.347·22-s − 3.96·23-s − 0.463·24-s − 0.487·26-s − 5.58·27-s + 1.99·28-s + 3.29·29-s + 10.0·31-s + 1.07·32-s − 4.94·33-s + 0.454·34-s + ⋯ |
| L(s) = 1 | + 0.0638·2-s + 0.742·3-s − 0.995·4-s + 0.0474·6-s − 0.377·7-s − 0.127·8-s − 0.449·9-s − 1.15·11-s − 0.739·12-s − 1.49·13-s − 0.0241·14-s + 0.987·16-s + 1.21·17-s − 0.0286·18-s + 0.229·19-s − 0.280·21-s − 0.0740·22-s − 0.827·23-s − 0.0946·24-s − 0.0955·26-s − 1.07·27-s + 0.376·28-s + 0.612·29-s + 1.81·31-s + 0.190·32-s − 0.860·33-s + 0.0779·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206878811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206878811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0903T + 2T^{2} \) |
| 3 | \( 1 - 1.28T + 3T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.63T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 3.49T + 73T^{2} \) |
| 79 | \( 1 - 3.22T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
| 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
−8.323029029064266280522414310711, −8.171473623171342095636897810349, −7.45614937122831451062978961982, −6.30550053912041845241646104724, −5.33050624706095629007337442839, −4.92769335164903701212022166751, −3.83713506873554700286965886375, −2.99407529075019634183731951074, −2.38326645837094592367399006627, −0.60191257831182200909378466425,
0.60191257831182200909378466425, 2.38326645837094592367399006627, 2.99407529075019634183731951074, 3.83713506873554700286965886375, 4.92769335164903701212022166751, 5.33050624706095629007337442839, 6.30550053912041845241646104724, 7.45614937122831451062978961982, 8.171473623171342095636897810349, 8.323029029064266280522414310711
