| L(s) = 1 | − 2.58·2-s + 1.40·3-s + 4.67·4-s − 2.16·5-s − 3.63·6-s + 7-s − 6.91·8-s − 1.01·9-s + 5.60·10-s − 3.16·11-s + 6.58·12-s + 1.21·13-s − 2.58·14-s − 3.05·15-s + 8.51·16-s + 4.73·17-s + 2.63·18-s − 4.76·19-s − 10.1·20-s + 1.40·21-s + 8.18·22-s − 5.72·23-s − 9.73·24-s − 0.298·25-s − 3.14·26-s − 5.65·27-s + 4.67·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 0.812·3-s + 2.33·4-s − 0.969·5-s − 1.48·6-s + 0.377·7-s − 2.44·8-s − 0.339·9-s + 1.77·10-s − 0.955·11-s + 1.90·12-s + 0.337·13-s − 0.690·14-s − 0.787·15-s + 2.12·16-s + 1.14·17-s + 0.620·18-s − 1.09·19-s − 2.26·20-s + 0.307·21-s + 1.74·22-s − 1.19·23-s − 1.98·24-s − 0.0597·25-s − 0.616·26-s − 1.08·27-s + 0.883·28-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
| \(L(1)\) |
\(\approx\) |
\(0.4513515125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4513515125\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 + 8.90T + 29T^{2} \) |
| 31 | \( 1 + 3.13T + 31T^{2} \) |
| 37 | \( 1 + 1.91T + 37T^{2} \) |
| 41 | \( 1 - 5.76T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 0.623T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 7.16T + 71T^{2} \) |
| 73 | \( 1 - 6.07T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−8.185678402830385742320856609403, −7.57185233106618622603110893084, −7.42418571041647208902462583103, −6.12550031646401640951918597540, −5.51975319135424949630137634406, −4.08148800541282380965825080334, −3.39077190844272937066673370164, −2.42120429573315041821073719024, −1.80867196685064658621809795580, −0.42525747926179960874644208573,
0.42525747926179960874644208573, 1.80867196685064658621809795580, 2.42120429573315041821073719024, 3.39077190844272937066673370164, 4.08148800541282380965825080334, 5.51975319135424949630137634406, 6.12550031646401640951918597540, 7.42418571041647208902462583103, 7.57185233106618622603110893084, 8.185678402830385742320856609403
