| L(s) = 1 | − 2-s + 4-s + 5-s + 3.08·7-s − 8-s − 10-s + 6.46·11-s + 3.95·13-s − 3.08·14-s + 16-s + 3.43·17-s + 3.08·19-s + 20-s − 6.46·22-s + 23-s + 25-s − 3.95·26-s + 3.08·28-s − 0.863·29-s − 5.95·31-s − 32-s − 3.43·34-s + 3.08·35-s − 7.03·37-s − 3.08·38-s − 40-s − 5.60·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.16·7-s − 0.353·8-s − 0.316·10-s + 1.95·11-s + 1.09·13-s − 0.825·14-s + 0.250·16-s + 0.832·17-s + 0.708·19-s + 0.223·20-s − 1.37·22-s + 0.208·23-s + 0.200·25-s − 0.774·26-s + 0.583·28-s − 0.160·29-s − 1.06·31-s − 0.176·32-s − 0.588·34-s + 0.521·35-s − 1.15·37-s − 0.500·38-s − 0.158·40-s − 0.875·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014855758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014855758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 29 | \( 1 + 0.863T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 + 7.03T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 3.90T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 2.56T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−9.072461687750138940451419152941, −8.525291059385044318268314902813, −7.64505031620058481563203189489, −6.87976357111305687994787129829, −6.04544570229625922977554260925, −5.29149124876157713310563113285, −4.11139061853621326477343850955, −3.25554100253115218425793341542, −1.62531890434199560998945431872, −1.29526058308459081231307607058,
1.29526058308459081231307607058, 1.62531890434199560998945431872, 3.25554100253115218425793341542, 4.11139061853621326477343850955, 5.29149124876157713310563113285, 6.04544570229625922977554260925, 6.87976357111305687994787129829, 7.64505031620058481563203189489, 8.525291059385044318268314902813, 9.072461687750138940451419152941
