L(s) = 1 | − 2-s − 3-s + 4-s + 1.11·5-s + 6-s + 0.414·7-s − 8-s + 9-s − 1.11·10-s + 1.07·11-s − 12-s + 4.69·13-s − 0.414·14-s − 1.11·15-s + 16-s − 2.07·17-s − 18-s − 19-s + 1.11·20-s − 0.414·21-s − 1.07·22-s + 0.442·23-s + 24-s − 3.74·25-s − 4.69·26-s − 27-s + 0.414·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.500·5-s + 0.408·6-s + 0.156·7-s − 0.353·8-s + 0.333·9-s − 0.354·10-s + 0.323·11-s − 0.288·12-s + 1.30·13-s − 0.110·14-s − 0.289·15-s + 0.250·16-s − 0.504·17-s − 0.235·18-s − 0.229·19-s + 0.250·20-s − 0.0904·21-s − 0.229·22-s + 0.0922·23-s + 0.204·24-s − 0.749·25-s − 0.921·26-s − 0.192·27-s + 0.0783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 - 1.11T + 5T^{2} \) |
| 7 | \( 1 - 0.414T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 23 | \( 1 - 0.442T + 23T^{2} \) |
| 29 | \( 1 + 5.42T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 + 6.51T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 - 1.28T + 61T^{2} \) |
| 67 | \( 1 + 0.633T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 0.429T + 73T^{2} \) |
| 79 | \( 1 + 4.95T + 79T^{2} \) |
| 83 | \( 1 - 1.90T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.81709381647979071707442340991, −6.87487728235781384856540194131, −6.37528518455242515637822985918, −5.79975334451199473487563313551, −4.96951381945429233890139956313, −4.02156848034438450402514522093, −3.17681756196313774934270219479, −1.91986902625605130382970634790, −1.34574726537964986671072293595, 0,
1.34574726537964986671072293595, 1.91986902625605130382970634790, 3.17681756196313774934270219479, 4.02156848034438450402514522093, 4.96951381945429233890139956313, 5.79975334451199473487563313551, 6.37528518455242515637822985918, 6.87487728235781384856540194131, 7.81709381647979071707442340991