L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 0.925·11-s − 12-s − 4.64·13-s + 14-s − 15-s + 16-s − 3.72·17-s + 18-s − 1.72·19-s + 20-s − 21-s − 0.925·22-s − 23-s − 24-s + 25-s − 4.64·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.278·11-s − 0.288·12-s − 1.28·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.902·17-s + 0.235·18-s − 0.394·19-s + 0.223·20-s − 0.218·21-s − 0.197·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s − 0.911·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.627526805\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.627526805\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 0.925T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 - 9.44T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 - 9.44T + 53T^{2} \) |
| 59 | \( 1 + 7.44T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 0.925T + 67T^{2} \) |
| 71 | \( 1 - 6.77T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 6.79T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.039013622218953728638447433406, −7.44106880794741231061886725057, −6.62083898252875593960871916940, −6.05569991959587379452565212302, −5.27659430406985619739790900959, −4.61443574394053645775748783633, −4.10178960771677848846864564130, −2.62961668710100263802782099444, −2.24219600718194982659014658916, −0.816445402049001905717273919158,
0.816445402049001905717273919158, 2.24219600718194982659014658916, 2.62961668710100263802782099444, 4.10178960771677848846864564130, 4.61443574394053645775748783633, 5.27659430406985619739790900959, 6.05569991959587379452565212302, 6.62083898252875593960871916940, 7.44106880794741231061886725057, 8.039013622218953728638447433406