| L(s) = 1 | + 2-s − 2.37·3-s + 4-s + 0.983·5-s − 2.37·6-s − 3.11·7-s + 8-s + 2.65·9-s + 0.983·10-s + 3.12·11-s − 2.37·12-s − 6.14·13-s − 3.11·14-s − 2.33·15-s + 16-s + 2.39·17-s + 2.65·18-s + 8.17·19-s + 0.983·20-s + 7.40·21-s + 3.12·22-s + 23-s − 2.37·24-s − 4.03·25-s − 6.14·26-s + 0.813·27-s − 3.11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.37·3-s + 0.5·4-s + 0.439·5-s − 0.971·6-s − 1.17·7-s + 0.353·8-s + 0.885·9-s + 0.310·10-s + 0.943·11-s − 0.686·12-s − 1.70·13-s − 0.831·14-s − 0.603·15-s + 0.250·16-s + 0.580·17-s + 0.626·18-s + 1.87·19-s + 0.219·20-s + 1.61·21-s + 0.666·22-s + 0.208·23-s − 0.485·24-s − 0.806·25-s − 1.20·26-s + 0.156·27-s − 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496471427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496471427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 0.983T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 - 8.17T + 19T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 0.415T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 4.37T + 67T^{2} \) |
| 71 | \( 1 - 0.807T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 3.21T + 79T^{2} \) |
| 83 | \( 1 - 9.74T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 3.00T + 97T^{2} \) |
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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
−9.869423164297944275596860614727, −9.184942706204372947698758047183, −7.43982195042548975734475428123, −6.98268321171158951559469528020, −6.04542130954234206915695814184, −5.57014492910346178447650608939, −4.77123225123700253958940646098, −3.63588340105450468174686783723, −2.55811707414204038184582213038, −0.862845583331986098621430322804,
0.862845583331986098621430322804, 2.55811707414204038184582213038, 3.63588340105450468174686783723, 4.77123225123700253958940646098, 5.57014492910346178447650608939, 6.04542130954234206915695814184, 6.98268321171158951559469528020, 7.43982195042548975734475428123, 9.184942706204372947698758047183, 9.869423164297944275596860614727
