| L(s) = 1 | − 1.87·2-s + 1.52·4-s + 1.30·5-s − 1.71·7-s + 0.896·8-s − 2.45·10-s + 5.30·11-s + 2.18·13-s + 3.21·14-s − 4.72·16-s − 0.392·17-s + 0.498·19-s + 1.98·20-s − 9.96·22-s − 8.33·23-s − 3.29·25-s − 4.10·26-s − 2.60·28-s + 4.50·29-s − 2.96·31-s + 7.07·32-s + 0.737·34-s − 2.23·35-s + 3.06·37-s − 0.936·38-s + 1.17·40-s + 10.0·41-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.761·4-s + 0.584·5-s − 0.647·7-s + 0.317·8-s − 0.775·10-s + 1.60·11-s + 0.606·13-s + 0.859·14-s − 1.18·16-s − 0.0952·17-s + 0.114·19-s + 0.444·20-s − 2.12·22-s − 1.73·23-s − 0.658·25-s − 0.805·26-s − 0.493·28-s + 0.837·29-s − 0.532·31-s + 1.25·32-s + 0.126·34-s − 0.378·35-s + 0.504·37-s − 0.151·38-s + 0.185·40-s + 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.053595735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.053595735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 + 0.392T + 17T^{2} \) |
| 19 | \( 1 - 0.498T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 + 0.714T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 + 0.889T + 61T^{2} \) |
| 67 | \( 1 - 4.67T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 + 7.79T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.437749189516572137022744385830, −7.69211561757342423646899703787, −6.90430971744185218637574580933, −6.22401818388874112401613957220, −5.75465861790173024066089363118, −4.27910550487585107486746106591, −3.84600308691018432798155816221, −2.48522550475514192127359838615, −1.62438316081586730440505620050, −0.72674469714753271782643944771,
0.72674469714753271782643944771, 1.62438316081586730440505620050, 2.48522550475514192127359838615, 3.84600308691018432798155816221, 4.27910550487585107486746106591, 5.75465861790173024066089363118, 6.22401818388874112401613957220, 6.90430971744185218637574580933, 7.69211561757342423646899703787, 8.437749189516572137022744385830
