| L(s) = 1 | − 2.78·2-s + 3.33·3-s + 5.74·4-s + 2.68·5-s − 9.29·6-s + 7-s − 10.4·8-s + 8.15·9-s − 7.48·10-s − 2.52·11-s + 19.1·12-s − 4.90·13-s − 2.78·14-s + 8.97·15-s + 17.5·16-s − 0.898·17-s − 22.6·18-s + 1.96·19-s + 15.4·20-s + 3.33·21-s + 7.03·22-s − 7.40·23-s − 34.8·24-s + 2.22·25-s + 13.6·26-s + 17.2·27-s + 5.74·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 1.92·3-s + 2.87·4-s + 1.20·5-s − 3.79·6-s + 0.377·7-s − 3.68·8-s + 2.71·9-s − 2.36·10-s − 0.761·11-s + 5.54·12-s − 1.36·13-s − 0.743·14-s + 2.31·15-s + 4.38·16-s − 0.217·17-s − 5.34·18-s + 0.450·19-s + 3.45·20-s + 0.728·21-s + 1.49·22-s − 1.54·23-s − 7.11·24-s + 0.444·25-s + 2.67·26-s + 3.31·27-s + 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.297804870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.297804870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 - 3.33T + 3T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 + 0.898T + 17T^{2} \) |
| 19 | \( 1 - 1.96T + 19T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 5.35T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 2.64T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 3.69T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 4.59T + 71T^{2} \) |
| 73 | \( 1 - 0.997T + 73T^{2} \) |
| 79 | \( 1 + 2.20T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 + 0.258T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.128357199798694658768044109615, −7.76760940802849719873048938290, −7.16893941700254570787321087489, −6.39530529133980573133061341195, −5.45341867227365169591429770709, −4.17803426184495005866367101149, −2.80553779171555684390333451861, −2.43815530836926588567288662201, −2.02995706288325383374471979200, −0.983210509569960461624709359181,
0.983210509569960461624709359181, 2.02995706288325383374471979200, 2.43815530836926588567288662201, 2.80553779171555684390333451861, 4.17803426184495005866367101149, 5.45341867227365169591429770709, 6.39530529133980573133061341195, 7.16893941700254570787321087489, 7.76760940802849719873048938290, 8.128357199798694658768044109615
