L(s) = 1 | + 1.11·2-s + 3-s − 0.761·4-s − 3.97·5-s + 1.11·6-s − 2.83·7-s − 3.07·8-s + 9-s − 4.42·10-s − 6.16·11-s − 0.761·12-s − 13-s − 3.14·14-s − 3.97·15-s − 1.89·16-s − 4.78·17-s + 1.11·18-s − 2.58·19-s + 3.03·20-s − 2.83·21-s − 6.85·22-s + 2.04·23-s − 3.07·24-s + 10.8·25-s − 1.11·26-s + 27-s + 2.15·28-s + ⋯ |
L(s) = 1 | + 0.786·2-s + 0.577·3-s − 0.380·4-s − 1.77·5-s + 0.454·6-s − 1.06·7-s − 1.08·8-s + 0.333·9-s − 1.39·10-s − 1.85·11-s − 0.219·12-s − 0.277·13-s − 0.841·14-s − 1.02·15-s − 0.474·16-s − 1.15·17-s + 0.262·18-s − 0.592·19-s + 0.677·20-s − 0.617·21-s − 1.46·22-s + 0.427·23-s − 0.627·24-s + 2.16·25-s − 0.218·26-s + 0.192·27-s + 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3784839443\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3784839443\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 1.11T + 2T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 + 2.83T + 7T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 + 0.839T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 0.205T + 43T^{2} \) |
| 47 | \( 1 + 5.44T + 47T^{2} \) |
| 53 | \( 1 + 9.08T + 53T^{2} \) |
| 59 | \( 1 + 9.01T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 - 5.27T + 67T^{2} \) |
| 71 | \( 1 + 0.585T + 71T^{2} \) |
| 73 | \( 1 + 2.33T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 0.848T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 6.93T + 97T^{2} \) |
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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
−8.298486997120775338074685978524, −7.83799214495329608136779370726, −7.00827112464371152977900130222, −6.28379362690076673703173255331, −5.10752124396372681302764213654, −4.54413726442523808908142732279, −3.88406432762897626733270463415, −3.04348115742056322689843567241, −2.65544087226839448471129472167, −0.28058185501594529184114611868,
0.28058185501594529184114611868, 2.65544087226839448471129472167, 3.04348115742056322689843567241, 3.88406432762897626733270463415, 4.54413726442523808908142732279, 5.10752124396372681302764213654, 6.28379362690076673703173255331, 7.00827112464371152977900130222, 7.83799214495329608136779370726, 8.298486997120775338074685978524