| L(s) = 1 | + 2.34·3-s − 3.81·5-s − 0.407·7-s + 2.51·9-s + 4.85·11-s − 2.78·13-s − 8.94·15-s − 1.04·17-s + 2.71·19-s − 0.956·21-s − 8.67·23-s + 9.52·25-s − 1.14·27-s − 4.08·29-s + 0.325·31-s + 11.4·33-s + 1.55·35-s + 11.4·37-s − 6.54·39-s + 0.439·41-s − 9.74·43-s − 9.57·45-s + 4.85·47-s − 6.83·49-s − 2.46·51-s − 8.80·53-s − 18.5·55-s + ⋯ |
| L(s) = 1 | + 1.35·3-s − 1.70·5-s − 0.154·7-s + 0.837·9-s + 1.46·11-s − 0.773·13-s − 2.31·15-s − 0.254·17-s + 0.622·19-s − 0.208·21-s − 1.80·23-s + 1.90·25-s − 0.220·27-s − 0.759·29-s + 0.0584·31-s + 1.98·33-s + 0.262·35-s + 1.88·37-s − 1.04·39-s + 0.0686·41-s − 1.48·43-s − 1.42·45-s + 0.708·47-s − 0.976·49-s − 0.344·51-s − 1.20·53-s − 2.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 + 3.81T + 5T^{2} \) |
| 7 | \( 1 + 0.407T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 13 | \( 1 + 2.78T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 2.71T + 19T^{2} \) |
| 23 | \( 1 + 8.67T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 0.325T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 0.439T + 41T^{2} \) |
| 43 | \( 1 + 9.74T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 6.37T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 - 2.77T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−7.994037935151349139661849805888, −7.69979696282387216811114804089, −6.94004442839650220618005636946, −6.07208196815888007250097057486, −4.65830405606779159802024547359, −4.02811421613778133390050268574, −3.53227230922997156708043131728, −2.73194970994956366794305130973, −1.56331440241063995892356136996, 0,
1.56331440241063995892356136996, 2.73194970994956366794305130973, 3.53227230922997156708043131728, 4.02811421613778133390050268574, 4.65830405606779159802024547359, 6.07208196815888007250097057486, 6.94004442839650220618005636946, 7.69979696282387216811114804089, 7.994037935151349139661849805888
