L(s) = 1 | − 2-s + 2.60·3-s + 4-s − 1.20·5-s − 2.60·6-s + 4.34·7-s − 8-s + 3.77·9-s + 1.20·10-s + 4.29·11-s + 2.60·12-s + 2.62·13-s − 4.34·14-s − 3.12·15-s + 16-s + 6.73·17-s − 3.77·18-s + 3.92·19-s − 1.20·20-s + 11.3·21-s − 4.29·22-s + 23-s − 2.60·24-s − 3.55·25-s − 2.62·26-s + 2.00·27-s + 4.34·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.50·3-s + 0.5·4-s − 0.537·5-s − 1.06·6-s + 1.64·7-s − 0.353·8-s + 1.25·9-s + 0.379·10-s + 1.29·11-s + 0.751·12-s + 0.727·13-s − 1.16·14-s − 0.807·15-s + 0.250·16-s + 1.63·17-s − 0.888·18-s + 0.901·19-s − 0.268·20-s + 2.46·21-s − 0.916·22-s + 0.208·23-s − 0.531·24-s − 0.711·25-s − 0.514·26-s + 0.386·27-s + 0.821·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.560851182\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.560851182\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 3.92T + 19T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 - 2.83T + 41T^{2} \) |
| 43 | \( 1 - 7.42T + 43T^{2} \) |
| 47 | \( 1 - 4.72T + 47T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 0.687T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 + 7.50T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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
−8.093666975072689600940157024626, −7.56321849712796225478575913439, −7.30231197361569548238768379684, −5.99801724338404198429254359318, −5.20231100567691997988617389795, −3.95840773746027673405497017957, −3.71988201516830599870062459964, −2.70988246675683450284801044985, −1.56794958557665161785492460173, −1.25603152112178430048344015991,
1.25603152112178430048344015991, 1.56794958557665161785492460173, 2.70988246675683450284801044985, 3.71988201516830599870062459964, 3.95840773746027673405497017957, 5.20231100567691997988617389795, 5.99801724338404198429254359318, 7.30231197361569548238768379684, 7.56321849712796225478575913439, 8.093666975072689600940157024626