| L(s) = 1 | − 1.74·2-s + 0.598·3-s + 1.03·4-s − 1.04·6-s − 1.03·7-s + 1.68·8-s − 2.64·9-s + 0.0764·11-s + 0.620·12-s − 2.86·13-s + 1.80·14-s − 4.99·16-s + 4.60·18-s + 4.19·19-s − 0.621·21-s − 0.133·22-s + 5.95·23-s + 1.00·24-s + 4.98·26-s − 3.37·27-s − 1.07·28-s − 0.0884·29-s − 2.42·31-s + 5.34·32-s + 0.0457·33-s − 2.73·36-s − 9.51·37-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.345·3-s + 0.517·4-s − 0.425·6-s − 0.392·7-s + 0.594·8-s − 0.880·9-s + 0.0230·11-s + 0.178·12-s − 0.793·13-s + 0.483·14-s − 1.24·16-s + 1.08·18-s + 0.961·19-s − 0.135·21-s − 0.0284·22-s + 1.24·23-s + 0.205·24-s + 0.977·26-s − 0.650·27-s − 0.203·28-s − 0.0164·29-s − 0.435·31-s + 0.945·32-s + 0.00796·33-s − 0.455·36-s − 1.56·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 - 0.598T + 3T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 0.0764T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 + 0.0884T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 + 9.51T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 0.866T + 43T^{2} \) |
| 47 | \( 1 - 8.33T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 + 9.16T + 59T^{2} \) |
| 61 | \( 1 + 0.288T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 6.54T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 - 0.106T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−7.57509174528457394984270785460, −7.27754977926065818306541520581, −6.42285708455959258008478125668, −5.42408332677987315037096629999, −4.87515460727177752847276139194, −3.74729457157198604399940156233, −2.93528865007478104664663021416, −2.16695100387509996963540852323, −1.03099152099989864544856515130, 0,
1.03099152099989864544856515130, 2.16695100387509996963540852323, 2.93528865007478104664663021416, 3.74729457157198604399940156233, 4.87515460727177752847276139194, 5.42408332677987315037096629999, 6.42285708455959258008478125668, 7.27754977926065818306541520581, 7.57509174528457394984270785460
