| L(s) = 1 | + 2.03·2-s − 2.37·3-s + 2.12·4-s − 4.81·6-s − 5.03·7-s + 0.248·8-s + 2.61·9-s − 1.90·11-s − 5.03·12-s − 1.04·13-s − 10.2·14-s − 3.74·16-s + 17-s + 5.31·18-s + 3.31·19-s + 11.9·21-s − 3.87·22-s − 0.125·23-s − 0.588·24-s − 2.11·26-s + 0.904·27-s − 10.6·28-s − 5.56·29-s − 4.99·31-s − 8.09·32-s + 4.52·33-s + 2.03·34-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 1.36·3-s + 1.06·4-s − 1.96·6-s − 1.90·7-s + 0.0877·8-s + 0.872·9-s − 0.575·11-s − 1.45·12-s − 0.288·13-s − 2.72·14-s − 0.935·16-s + 0.242·17-s + 1.25·18-s + 0.761·19-s + 2.60·21-s − 0.825·22-s − 0.0262·23-s − 0.120·24-s − 0.414·26-s + 0.174·27-s − 2.01·28-s − 1.03·29-s − 0.897·31-s − 1.43·32-s + 0.787·33-s + 0.348·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 19 | \( 1 - 3.31T + 19T^{2} \) |
| 23 | \( 1 + 0.125T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 47 | \( 1 + 1.04T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 7.14T + 61T^{2} \) |
| 67 | \( 1 + 3.28T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 + 7.29T + 97T^{2} \) |
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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
−11.07326621373288690312676460997, −10.06095453146652913071519164657, −9.248765511384459495233527803327, −7.32351345978656237153008501324, −6.47723379136573475744573161141, −5.77437945986760715980437515358, −5.14425527419053531377254544037, −3.84008412583519439703624759400, −2.86537473873942570136796758259, 0,
2.86537473873942570136796758259, 3.84008412583519439703624759400, 5.14425527419053531377254544037, 5.77437945986760715980437515358, 6.47723379136573475744573161141, 7.32351345978656237153008501324, 9.248765511384459495233527803327, 10.06095453146652913071519164657, 11.07326621373288690312676460997
