L(s) = 1 | − 1.92·2-s − 3-s + 1.70·4-s − 5-s + 1.92·6-s − 7-s + 0.576·8-s + 9-s + 1.92·10-s − 5.42·11-s − 1.70·12-s + 2.22·13-s + 1.92·14-s + 15-s − 4.50·16-s − 2·17-s − 1.92·18-s + 6.85·19-s − 1.70·20-s + 21-s + 10.4·22-s − 23-s − 0.576·24-s + 25-s − 4.27·26-s − 27-s − 1.70·28-s + ⋯ |
L(s) = 1 | − 1.36·2-s − 0.577·3-s + 0.850·4-s − 0.447·5-s + 0.785·6-s − 0.377·7-s + 0.203·8-s + 0.333·9-s + 0.608·10-s − 1.63·11-s − 0.490·12-s + 0.616·13-s + 0.514·14-s + 0.258·15-s − 1.12·16-s − 0.485·17-s − 0.453·18-s + 1.57·19-s − 0.380·20-s + 0.218·21-s + 2.22·22-s − 0.208·23-s − 0.117·24-s + 0.200·25-s − 0.838·26-s − 0.192·27-s − 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 29 | \( 1 - 4.49T + 29T^{2} \) |
| 31 | \( 1 - 0.513T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 + 0.509T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 - 1.20T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 2.25T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 1.48T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
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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
−8.422906962517877834126632856019, −8.000515332010103752468841305645, −7.23064809501071904441687308271, −6.55500283724255144652814165170, −5.43363483300080064224737261016, −4.76874164883960347585013073173, −3.52892854823574043520495298055, −2.41912480535167910338672323361, −1.05079795616744285391470580377, 0,
1.05079795616744285391470580377, 2.41912480535167910338672323361, 3.52892854823574043520495298055, 4.76874164883960347585013073173, 5.43363483300080064224737261016, 6.55500283724255144652814165170, 7.23064809501071904441687308271, 8.000515332010103752468841305645, 8.422906962517877834126632856019