| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 2.61·5-s − 1.61·6-s + 7-s − 2.23·8-s + 9-s − 4.23·10-s − 2.23·11-s − 0.618·12-s − 6.85·13-s + 1.61·14-s + 2.61·15-s − 4.85·16-s + 3.47·17-s + 1.61·18-s − 3.76·19-s − 1.61·20-s − 21-s − 3.61·22-s + 23-s + 2.23·24-s + 1.85·25-s − 11.0·26-s − 27-s + 0.618·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 1.17·5-s − 0.660·6-s + 0.377·7-s − 0.790·8-s + 0.333·9-s − 1.33·10-s − 0.674·11-s − 0.178·12-s − 1.90·13-s + 0.432·14-s + 0.675·15-s − 1.21·16-s + 0.842·17-s + 0.381·18-s − 0.863·19-s − 0.361·20-s − 0.218·21-s − 0.771·22-s + 0.208·23-s + 0.456·24-s + 0.370·25-s − 2.17·26-s − 0.192·27-s + 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 6.85T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 0.763T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 - 2.09T + 61T^{2} \) |
| 67 | \( 1 + 0.145T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 9.94T + 83T^{2} \) |
| 89 | \( 1 + 0.673T + 89T^{2} \) |
| 97 | \( 1 - 5.18T + 97T^{2} \) |
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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
−10.79450593174039892510984598321, −9.893281986431813321615983652435, −8.546331384402012229223328991291, −7.60114981634264924556613022056, −6.75100769493312226531274994752, −5.36091861991242819198887299311, −4.83836721377746536790923230274, −3.92484120595962203859369866348, −2.66890882043305631932503004946, 0,
2.66890882043305631932503004946, 3.92484120595962203859369866348, 4.83836721377746536790923230274, 5.36091861991242819198887299311, 6.75100769493312226531274994752, 7.60114981634264924556613022056, 8.546331384402012229223328991291, 9.893281986431813321615983652435, 10.79450593174039892510984598321
