| L(s) = 1 | − 1.89·2-s + 1.58·4-s − 5-s + 1.67·7-s + 0.780·8-s + 1.89·10-s + 6.42·11-s + 13-s − 3.17·14-s − 4.65·16-s − 1.05·17-s − 3.65·19-s − 1.58·20-s − 12.1·22-s − 1.36·23-s + 25-s − 1.89·26-s + 2.66·28-s + 7.54·29-s − 6.86·31-s + 7.25·32-s + 2.00·34-s − 1.67·35-s + 1.00·37-s + 6.93·38-s − 0.780·40-s − 8.49·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.794·4-s − 0.447·5-s + 0.633·7-s + 0.275·8-s + 0.599·10-s + 1.93·11-s + 0.277·13-s − 0.848·14-s − 1.16·16-s − 0.256·17-s − 0.839·19-s − 0.355·20-s − 2.59·22-s − 0.283·23-s + 0.200·25-s − 0.371·26-s + 0.502·28-s + 1.40·29-s − 1.23·31-s + 1.28·32-s + 0.343·34-s − 0.283·35-s + 0.164·37-s + 1.12·38-s − 0.123·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 - 6.42T + 11T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 + 8.49T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 - 2.39T + 47T^{2} \) |
| 53 | \( 1 + 8.53T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 2.39T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 5.02T + 89T^{2} \) |
| 97 | \( 1 + 0.864T + 97T^{2} \) |
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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
−8.172064126478761626009147480959, −7.15923034833415032396972158183, −6.77034100879364629184363729423, −5.95603716434254816922966416799, −4.63169693084735186536472634452, −4.24962884364152051272794331861, −3.22407013433849289047353418008, −1.78743801790220303790803346445, −1.33263346597517831921468197925, 0,
1.33263346597517831921468197925, 1.78743801790220303790803346445, 3.22407013433849289047353418008, 4.24962884364152051272794331861, 4.63169693084735186536472634452, 5.95603716434254816922966416799, 6.77034100879364629184363729423, 7.15923034833415032396972158183, 8.172064126478761626009147480959
