L(s) = 1 | + 2.53·5-s − 7-s − 0.467·11-s + 5.82·13-s + 3.87·17-s + 2.18·19-s + 0.106·23-s + 1.41·25-s + 8.78·29-s + 7.68·31-s − 2.53·35-s − 7.68·37-s − 2.22·41-s − 1.22·43-s + 5.33·47-s + 49-s − 0.716·53-s − 1.18·55-s − 0.736·59-s + 0.958·61-s + 14.7·65-s + 9.63·67-s − 13.2·71-s − 10.2·73-s + 0.467·77-s + 12.6·79-s + 2.73·83-s + ⋯ |
L(s) = 1 | + 1.13·5-s − 0.377·7-s − 0.141·11-s + 1.61·13-s + 0.940·17-s + 0.501·19-s + 0.0221·23-s + 0.282·25-s + 1.63·29-s + 1.37·31-s − 0.428·35-s − 1.26·37-s − 0.347·41-s − 0.187·43-s + 0.777·47-s + 0.142·49-s − 0.0984·53-s − 0.159·55-s − 0.0958·59-s + 0.122·61-s + 1.82·65-s + 1.17·67-s − 1.57·71-s − 1.20·73-s + 0.0533·77-s + 1.42·79-s + 0.299·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.105478382\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.105478382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 2.53T + 5T^{2} \) |
| 11 | \( 1 + 0.467T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 - 0.106T + 23T^{2} \) |
| 29 | \( 1 - 8.78T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 + 0.716T + 53T^{2} \) |
| 59 | \( 1 + 0.736T + 59T^{2} \) |
| 61 | \( 1 - 0.958T + 61T^{2} \) |
| 67 | \( 1 - 9.63T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.81586846372024088728746033368, −6.80687196623650056496650641524, −6.34397593266677109621226637690, −5.72646314981267686523808582316, −5.17420839596661862399340767630, −4.19135645036486162806437656035, −3.31512700416462232078905957249, −2.71816713077920680324747703884, −1.60837501730279034657221610603, −0.927342309117779034536445153365,
0.927342309117779034536445153365, 1.60837501730279034657221610603, 2.71816713077920680324747703884, 3.31512700416462232078905957249, 4.19135645036486162806437656035, 5.17420839596661862399340767630, 5.72646314981267686523808582316, 6.34397593266677109621226637690, 6.80687196623650056496650641524, 7.81586846372024088728746033368