| L(s) = 1 | − 2·2-s − 2·3-s + 4·4-s + 12·5-s + 4·6-s − 7·7-s − 8·8-s − 23·9-s − 24·10-s − 48·11-s − 8·12-s + 14·14-s − 24·15-s + 16·16-s − 114·17-s + 46·18-s − 2·19-s + 48·20-s + 14·21-s + 96·22-s − 120·23-s + 16·24-s + 19·25-s + 100·27-s − 28·28-s − 54·29-s + 48·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 1/2·4-s + 1.07·5-s + 0.272·6-s − 0.377·7-s − 0.353·8-s − 0.851·9-s − 0.758·10-s − 1.31·11-s − 0.192·12-s + 0.267·14-s − 0.413·15-s + 1/4·16-s − 1.62·17-s + 0.602·18-s − 0.0241·19-s + 0.536·20-s + 0.145·21-s + 0.930·22-s − 1.08·23-s + 0.136·24-s + 0.151·25-s + 0.712·27-s − 0.188·28-s − 0.345·29-s + 0.292·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3393665913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3393665913\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 236 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 - 12 T + p^{3} T^{2} \) |
| 53 | \( 1 - 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 138 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 - 484 T + p^{3} T^{2} \) |
| 71 | \( 1 + 576 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 + 378 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1330 T + p^{3} T^{2} \) |
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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
−8.689918577341534601607368510722, −8.035383920032615569870445705122, −6.99753270803448174709063639141, −6.30590611123591386642937756740, −5.62062068375205612290022059866, −5.00284709574494067580903653811, −3.54992749581684870324602648826, −2.39959697319648438952443362485, −1.94899784658941992286394009427, −0.27103168103429384759058224596,
0.27103168103429384759058224596, 1.94899784658941992286394009427, 2.39959697319648438952443362485, 3.54992749581684870324602648826, 5.00284709574494067580903653811, 5.62062068375205612290022059866, 6.30590611123591386642937756740, 6.99753270803448174709063639141, 8.035383920032615569870445705122, 8.689918577341534601607368510722
