L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 0.801·7-s + 8-s + 9-s + 10-s − 3.04·11-s + 12-s − 0.801·14-s + 15-s + 16-s − 1.24·17-s + 18-s + 7.40·19-s + 20-s − 0.801·21-s − 3.04·22-s − 0.356·23-s + 24-s + 25-s + 27-s − 0.801·28-s + 0.307·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.303·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.919·11-s + 0.288·12-s − 0.214·14-s + 0.258·15-s + 0.250·16-s − 0.302·17-s + 0.235·18-s + 1.69·19-s + 0.223·20-s − 0.174·21-s − 0.650·22-s − 0.0744·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.151·28-s + 0.0571·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.216977937\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.216977937\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 0.801T + 7T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 23 | \( 1 + 0.356T + 23T^{2} \) |
| 29 | \( 1 - 0.307T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 7.29T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 - 6.54T + 43T^{2} \) |
| 47 | \( 1 - 0.603T + 47T^{2} \) |
| 53 | \( 1 + 6.29T + 53T^{2} \) |
| 59 | \( 1 - 3.11T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 5.96T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 3.93T + 83T^{2} \) |
| 89 | \( 1 + 8.05T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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
−8.031893611908137858807330313609, −7.54164386573055380415318714192, −6.75352527427356141610842105380, −5.96399434841614461075419568017, −5.27936816623326693360537572118, −4.58179429099176815965381596974, −3.62073527666662426948229648230, −2.85231478804974260955404782235, −2.27334829449510155506038780745, −1.00668811229990310442499686659,
1.00668811229990310442499686659, 2.27334829449510155506038780745, 2.85231478804974260955404782235, 3.62073527666662426948229648230, 4.58179429099176815965381596974, 5.27936816623326693360537572118, 5.96399434841614461075419568017, 6.75352527427356141610842105380, 7.54164386573055380415318714192, 8.031893611908137858807330313609