| L(s) = 1 | + 2-s + 1.79·3-s + 4-s + 3.79·5-s + 1.79·6-s − 4.79·7-s + 8-s + 0.208·9-s + 3.79·10-s + 11-s + 1.79·12-s + 1.20·13-s − 4.79·14-s + 6.79·15-s + 16-s − 7.58·17-s + 0.208·18-s + 19-s + 3.79·20-s − 8.58·21-s + 22-s − 1.58·23-s + 1.79·24-s + 9.37·25-s + 1.20·26-s − 5.00·27-s − 4.79·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.03·3-s + 0.5·4-s + 1.69·5-s + 0.731·6-s − 1.81·7-s + 0.353·8-s + 0.0695·9-s + 1.19·10-s + 0.301·11-s + 0.517·12-s + 0.335·13-s − 1.28·14-s + 1.75·15-s + 0.250·16-s − 1.83·17-s + 0.0491·18-s + 0.229·19-s + 0.847·20-s − 1.87·21-s + 0.213·22-s − 0.329·23-s + 0.365·24-s + 1.87·25-s + 0.237·26-s − 0.962·27-s − 0.905·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.966863533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.966863533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.79T + 5T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 0.791T + 41T^{2} \) |
| 43 | \( 1 - 7.37T + 43T^{2} \) |
| 47 | \( 1 - 9.16T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.58T + 73T^{2} \) |
| 79 | \( 1 + 5.58T + 79T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−11.10583277443275894614315838219, −10.11545094202045986032712995211, −9.226419795770132772882296649369, −8.959458917233339360022930203134, −7.18004497941309153151023509114, −6.27466227308965272853897334351, −5.74350977419233896713565907544, −4.04630285366451039518773781514, −2.90432409493106914981412676170, −2.16056460916487198026875068826,
2.16056460916487198026875068826, 2.90432409493106914981412676170, 4.04630285366451039518773781514, 5.74350977419233896713565907544, 6.27466227308965272853897334351, 7.18004497941309153151023509114, 8.959458917233339360022930203134, 9.226419795770132772882296649369, 10.11545094202045986032712995211, 11.10583277443275894614315838219
