L(s) = 1 | − 0.869·2-s − 3-s − 1.24·4-s − 3.05·5-s + 0.869·6-s + 7-s + 2.81·8-s + 9-s + 2.65·10-s + 0.712·11-s + 1.24·12-s − 4.02·13-s − 0.869·14-s + 3.05·15-s + 0.0388·16-s + 1.21·17-s − 0.869·18-s − 3.38·19-s + 3.79·20-s − 21-s − 0.618·22-s + 2.85·23-s − 2.81·24-s + 4.31·25-s + 3.49·26-s − 27-s − 1.24·28-s + ⋯ |
L(s) = 1 | − 0.614·2-s − 0.577·3-s − 0.622·4-s − 1.36·5-s + 0.354·6-s + 0.377·7-s + 0.996·8-s + 0.333·9-s + 0.838·10-s + 0.214·11-s + 0.359·12-s − 1.11·13-s − 0.232·14-s + 0.787·15-s + 0.00971·16-s + 0.295·17-s − 0.204·18-s − 0.775·19-s + 0.849·20-s − 0.218·21-s − 0.131·22-s + 0.595·23-s − 0.575·24-s + 0.862·25-s + 0.685·26-s − 0.192·27-s − 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3425662585\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3425662585\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 + 0.869T + 2T^{2} \) |
| 5 | \( 1 + 3.05T + 5T^{2} \) |
| 11 | \( 1 - 0.712T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 17 | \( 1 - 1.21T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 - 5.92T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 4.36T + 59T^{2} \) |
| 61 | \( 1 + 1.75T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 0.646T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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
−7.911526767288661842825181766518, −7.32037522902392784317245656231, −6.74446775873039616672025951411, −5.55443617237279581910900725823, −4.98829689493909773349671230318, −4.18813875608804589752298744123, −3.89944935982196617370884360039, −2.61181199141267817665084514759, −1.37171900936311466965882122657, −0.35877105121640986116297723178,
0.35877105121640986116297723178, 1.37171900936311466965882122657, 2.61181199141267817665084514759, 3.89944935982196617370884360039, 4.18813875608804589752298744123, 4.98829689493909773349671230318, 5.55443617237279581910900725823, 6.74446775873039616672025951411, 7.32037522902392784317245656231, 7.911526767288661842825181766518