L(s) = 1 | − 0.439·2-s + 3-s − 1.80·4-s + 5-s − 0.439·6-s − 0.915·7-s + 1.67·8-s + 9-s − 0.439·10-s + 0.783·11-s − 1.80·12-s + 0.245·13-s + 0.402·14-s + 15-s + 2.87·16-s + 1.06·17-s − 0.439·18-s + 0.848·19-s − 1.80·20-s − 0.915·21-s − 0.344·22-s − 5.37·23-s + 1.67·24-s + 25-s − 0.108·26-s + 27-s + 1.65·28-s + ⋯ |
L(s) = 1 | − 0.310·2-s + 0.577·3-s − 0.903·4-s + 0.447·5-s − 0.179·6-s − 0.345·7-s + 0.591·8-s + 0.333·9-s − 0.139·10-s + 0.236·11-s − 0.521·12-s + 0.0681·13-s + 0.107·14-s + 0.258·15-s + 0.719·16-s + 0.257·17-s − 0.103·18-s + 0.194·19-s − 0.403·20-s − 0.199·21-s − 0.0734·22-s − 1.12·23-s + 0.341·24-s + 0.200·25-s − 0.0211·26-s + 0.192·27-s + 0.312·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.794522968\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.794522968\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 0.439T + 2T^{2} \) |
| 7 | \( 1 + 0.915T + 7T^{2} \) |
| 11 | \( 1 - 0.783T + 11T^{2} \) |
| 13 | \( 1 - 0.245T + 13T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 - 0.848T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 - 5.35T + 59T^{2} \) |
| 61 | \( 1 - 4.20T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 - 6.13T + 73T^{2} \) |
| 79 | \( 1 + 9.10T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.275572606848009513308230315283, −7.64269312269723474579596585032, −6.57719212223258701953552303988, −6.11007519768832013435264886937, −5.00569886192087918474536083172, −4.49591247367509711131782979638, −3.58390390755908273065565419746, −2.85743951254226237406304475963, −1.74525209887592522567506920485, −0.74884726967266115770281726614,
0.74884726967266115770281726614, 1.74525209887592522567506920485, 2.85743951254226237406304475963, 3.58390390755908273065565419746, 4.49591247367509711131782979638, 5.00569886192087918474536083172, 6.11007519768832013435264886937, 6.57719212223258701953552303988, 7.64269312269723474579596585032, 8.275572606848009513308230315283