| L(s) = 1 | + 0.806·3-s − 0.193·5-s + 0.518·7-s − 2.35·9-s + 2.67·11-s + 1.96·13-s − 0.156·15-s − 0.806·17-s + 7.11·19-s + 0.418·21-s + 3.32·23-s − 4.96·25-s − 4.31·27-s + 7.89·29-s + 9.11·31-s + 2.15·33-s − 0.100·35-s − 5.76·37-s + 1.58·39-s + 7.70·41-s − 1.76·43-s + 0.455·45-s + 8.96·47-s − 6.73·49-s − 0.649·51-s − 5.66·53-s − 0.518·55-s + ⋯ |
| L(s) = 1 | + 0.465·3-s − 0.0867·5-s + 0.196·7-s − 0.783·9-s + 0.806·11-s + 0.544·13-s − 0.0403·15-s − 0.195·17-s + 1.63·19-s + 0.0912·21-s + 0.693·23-s − 0.992·25-s − 0.829·27-s + 1.46·29-s + 1.63·31-s + 0.375·33-s − 0.0170·35-s − 0.948·37-s + 0.253·39-s + 1.20·41-s − 0.269·43-s + 0.0679·45-s + 1.30·47-s − 0.961·49-s − 0.0909·51-s − 0.777·53-s − 0.0699·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.941874634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.941874634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 - 0.806T + 3T^{2} \) |
| 5 | \( 1 + 0.193T + 5T^{2} \) |
| 7 | \( 1 - 0.518T + 7T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 - 1.96T + 13T^{2} \) |
| 17 | \( 1 + 0.806T + 17T^{2} \) |
| 19 | \( 1 - 7.11T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 - 9.11T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 - 8.96T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 0.100T + 79T^{2} \) |
| 83 | \( 1 + 3.22T + 83T^{2} \) |
| 89 | \( 1 + 3.66T + 89T^{2} \) |
| 97 | \( 1 + 6.77T + 97T^{2} \) |
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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
−9.842216044912893808065484610626, −9.108046263799627164248344940249, −8.395910722237002436075492586306, −7.64271135833722255120148002777, −6.59127712889514036005031749972, −5.74583067031733015092869744240, −4.67605540288436608669937450354, −3.55726388552812243331815259923, −2.69847364218250534938970989842, −1.16402743028815860142029599059,
1.16402743028815860142029599059, 2.69847364218250534938970989842, 3.55726388552812243331815259923, 4.67605540288436608669937450354, 5.74583067031733015092869744240, 6.59127712889514036005031749972, 7.64271135833722255120148002777, 8.395910722237002436075492586306, 9.108046263799627164248344940249, 9.842216044912893808065484610626
