| L(s) = 1 | − 2.67·2-s + 2.10·3-s + 5.15·4-s − 5-s − 5.64·6-s + 2.87·7-s − 8.44·8-s + 1.45·9-s + 2.67·10-s − 4.04·11-s + 10.8·12-s − 3.14·13-s − 7.69·14-s − 2.10·15-s + 12.2·16-s − 0.961·17-s − 3.88·18-s + 0.696·19-s − 5.15·20-s + 6.06·21-s + 10.8·22-s + 0.357·23-s − 17.8·24-s + 25-s + 8.41·26-s − 3.26·27-s + 14.8·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 1.21·3-s + 2.57·4-s − 0.447·5-s − 2.30·6-s + 1.08·7-s − 2.98·8-s + 0.484·9-s + 0.846·10-s − 1.21·11-s + 3.14·12-s − 0.871·13-s − 2.05·14-s − 0.544·15-s + 3.07·16-s − 0.233·17-s − 0.915·18-s + 0.159·19-s − 1.15·20-s + 1.32·21-s + 2.30·22-s + 0.0746·23-s − 3.63·24-s + 0.200·25-s + 1.64·26-s − 0.628·27-s + 2.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 - 2.10T + 3T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 + 0.961T + 17T^{2} \) |
| 19 | \( 1 - 0.696T + 19T^{2} \) |
| 23 | \( 1 - 0.357T + 23T^{2} \) |
| 29 | \( 1 + 6.80T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 + 3.91T + 37T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 - 0.470T + 43T^{2} \) |
| 47 | \( 1 - 5.50T + 47T^{2} \) |
| 53 | \( 1 + 0.749T + 53T^{2} \) |
| 59 | \( 1 + 4.44T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 - 7.43T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.808820296422571924334497889608, −7.894254620638412238033013460302, −7.77304227059633480814779819346, −7.14311125932742666167172155812, −5.79197108106619971145576050581, −4.64059444277824247891160873959, −3.15290556309274943215112785613, −2.44229701140094375253980051094, −1.62901646369115089481322100718, 0,
1.62901646369115089481322100718, 2.44229701140094375253980051094, 3.15290556309274943215112785613, 4.64059444277824247891160873959, 5.79197108106619971145576050581, 7.14311125932742666167172155812, 7.77304227059633480814779819346, 7.894254620638412238033013460302, 8.808820296422571924334497889608
