| L(s) = 1 | − 1.74·2-s − 2.85·3-s + 1.03·4-s − 1.07·5-s + 4.97·6-s + 2.85·7-s + 1.68·8-s + 5.17·9-s + 1.86·10-s − 11-s − 2.95·12-s + 0.0989·13-s − 4.96·14-s + 3.06·15-s − 4.99·16-s + 17-s − 9.00·18-s + 0.192·19-s − 1.10·20-s − 8.15·21-s + 1.74·22-s + 5.90·23-s − 4.81·24-s − 3.84·25-s − 0.172·26-s − 6.20·27-s + 2.94·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 1.65·3-s + 0.516·4-s − 0.479·5-s + 2.03·6-s + 1.07·7-s + 0.595·8-s + 1.72·9-s + 0.590·10-s − 0.301·11-s − 0.851·12-s + 0.0274·13-s − 1.32·14-s + 0.791·15-s − 1.24·16-s + 0.242·17-s − 2.12·18-s + 0.0441·19-s − 0.247·20-s − 1.77·21-s + 0.371·22-s + 1.23·23-s − 0.983·24-s − 0.769·25-s − 0.0337·26-s − 1.19·27-s + 0.556·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4195723368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4195723368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 - 0.0989T + 13T^{2} \) |
| 19 | \( 1 - 0.192T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 + 7.37T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 - 1.29T + 37T^{2} \) |
| 41 | \( 1 - 4.87T + 41T^{2} \) |
| 47 | \( 1 - 1.86T + 47T^{2} \) |
| 53 | \( 1 - 8.96T + 53T^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 - 4.82T + 61T^{2} \) |
| 67 | \( 1 + 3.54T + 67T^{2} \) |
| 71 | \( 1 - 0.305T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 9.96T + 79T^{2} \) |
| 83 | \( 1 + 1.72T + 83T^{2} \) |
| 89 | \( 1 - 3.99T + 89T^{2} \) |
| 97 | \( 1 - 5.64T + 97T^{2} \) |
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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
−7.58118465934304583352431586888, −7.49640508014089234071640915384, −6.60635002500559846547404070480, −5.68663923646543910788261840587, −5.12763928417868535451210240088, −4.54490197044011010223298145560, −3.72811966566619225653856761566, −2.14069101230844569959054593162, −1.27336050345294759233051049099, −0.48855633383406667747980291600,
0.48855633383406667747980291600, 1.27336050345294759233051049099, 2.14069101230844569959054593162, 3.72811966566619225653856761566, 4.54490197044011010223298145560, 5.12763928417868535451210240088, 5.68663923646543910788261840587, 6.60635002500559846547404070480, 7.49640508014089234071640915384, 7.58118465934304583352431586888
