| L(s) = 1 | − 2-s − 3-s + 4-s − 2.70·5-s + 6-s − 8-s + 9-s + 2.70·10-s − 12-s + 0.701·13-s + 2.70·15-s + 16-s − 18-s − 7.40·19-s − 2.70·20-s − 23-s + 24-s + 2.29·25-s − 0.701·26-s − 27-s + 6.70·29-s − 2.70·30-s − 6·31-s − 32-s + 36-s − 2.70·37-s + 7.40·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.20·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.854·10-s − 0.288·12-s + 0.194·13-s + 0.697·15-s + 0.250·16-s − 0.235·18-s − 1.69·19-s − 0.604·20-s − 0.208·23-s + 0.204·24-s + 0.459·25-s − 0.137·26-s − 0.192·27-s + 1.24·29-s − 0.493·30-s − 1.07·31-s − 0.176·32-s + 0.166·36-s − 0.444·37-s + 1.20·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4060313413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4060313413\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.701T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.40T + 19T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 43 | \( 1 + 0.701T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 + 7.40T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 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
−8.104604702586528655906445633738, −7.29569932191362623130044007211, −6.71499611324612220429945360873, −6.07756409692961215654746137915, −5.14341383306162445390040615121, −4.26183198850873704387624631815, −3.72988250992142800093473741792, −2.63960247895371269782993674535, −1.57682909990166113307177242521, −0.37753831988603133028920515211,
0.37753831988603133028920515211, 1.57682909990166113307177242521, 2.63960247895371269782993674535, 3.72988250992142800093473741792, 4.26183198850873704387624631815, 5.14341383306162445390040615121, 6.07756409692961215654746137915, 6.71499611324612220429945360873, 7.29569932191362623130044007211, 8.104604702586528655906445633738
