| L(s) = 1 | + 2-s − 0.414·3-s + 4-s − 1.82·5-s − 0.414·6-s + 8-s − 2.82·9-s − 1.82·10-s − 6.24·11-s − 0.414·12-s + 0.585·13-s + 0.757·15-s + 16-s + 3.82·17-s − 2.82·18-s + 2.82·19-s − 1.82·20-s − 6.24·22-s + 5.65·23-s − 0.414·24-s − 1.65·25-s + 0.585·26-s + 2.41·27-s − 5·29-s + 0.757·30-s − 5.58·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.239·3-s + 0.5·4-s − 0.817·5-s − 0.169·6-s + 0.353·8-s − 0.942·9-s − 0.578·10-s − 1.88·11-s − 0.119·12-s + 0.162·13-s + 0.195·15-s + 0.250·16-s + 0.928·17-s − 0.666·18-s + 0.648·19-s − 0.408·20-s − 1.33·22-s + 1.17·23-s − 0.0845·24-s − 0.331·25-s + 0.114·26-s + 0.464·27-s − 0.928·29-s + 0.138·30-s − 1.00·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.665628600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665628600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 6.58T + 37T^{2} \) |
| 43 | \( 1 - 1.24T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 7.48T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 7.17T + 67T^{2} \) |
| 71 | \( 1 - 8.89T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 - 0.171T + 97T^{2} \) |
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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
−8.149498604435664574831816036032, −7.66442045379862573925047676400, −7.12908054837294707445034687046, −5.87159748867134372785048940290, −5.45895702909368595861200797339, −4.86475501906090255890633155631, −3.73378178317994568631739624083, −3.10513150521653224350552901401, −2.31304907826566716748110824470, −0.64326099353693652044440518949,
0.64326099353693652044440518949, 2.31304907826566716748110824470, 3.10513150521653224350552901401, 3.73378178317994568631739624083, 4.86475501906090255890633155631, 5.45895702909368595861200797339, 5.87159748867134372785048940290, 7.12908054837294707445034687046, 7.66442045379862573925047676400, 8.149498604435664574831816036032
