| L(s) = 1 | − 0.0191·2-s + 1.86·3-s − 1.99·4-s − 3.05·5-s − 0.0357·6-s − 1.55·7-s + 0.0766·8-s + 0.483·9-s + 0.0584·10-s + 1.27·11-s − 3.73·12-s + 3.24·13-s + 0.0297·14-s − 5.70·15-s + 3.99·16-s − 17-s − 0.00926·18-s + 6.72·19-s + 6.10·20-s − 2.90·21-s − 0.0245·22-s + 8.38·23-s + 0.142·24-s + 4.32·25-s − 0.0620·26-s − 4.69·27-s + 3.10·28-s + ⋯ |
| L(s) = 1 | − 0.0135·2-s + 1.07·3-s − 0.999·4-s − 1.36·5-s − 0.0145·6-s − 0.587·7-s + 0.0270·8-s + 0.161·9-s + 0.0184·10-s + 0.385·11-s − 1.07·12-s + 0.898·13-s + 0.00795·14-s − 1.47·15-s + 0.999·16-s − 0.242·17-s − 0.00218·18-s + 1.54·19-s + 1.36·20-s − 0.633·21-s − 0.00522·22-s + 1.74·23-s + 0.0291·24-s + 0.865·25-s − 0.0121·26-s − 0.903·27-s + 0.587·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.336762344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336762344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0191T + 2T^{2} \) |
| 3 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 + 3.05T + 5T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 - 8.38T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 - 0.0319T + 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 - 3.84T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 - 7.94T + 47T^{2} \) |
| 53 | \( 1 - 7.06T + 53T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 6.61T + 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 - 4.13T + 73T^{2} \) |
| 79 | \( 1 - 5.60T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 4.28T + 89T^{2} \) |
| 97 | \( 1 + 5.05T + 97T^{2} \) |
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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
−9.600928927443941380652828139596, −8.984440441951239345633161152675, −8.459511737740780038736754977658, −7.70968612642952122393325510817, −6.88064693565496084121204056311, −5.49267934642453316204379197613, −4.38697320982521939098341015593, −3.51513456689399452906626455199, −3.08901773356113560886770034835, −0.886058445952508242441234303273,
0.886058445952508242441234303273, 3.08901773356113560886770034835, 3.51513456689399452906626455199, 4.38697320982521939098341015593, 5.49267934642453316204379197613, 6.88064693565496084121204056311, 7.70968612642952122393325510817, 8.459511737740780038736754977658, 8.984440441951239345633161152675, 9.600928927443941380652828139596
