| L(s) = 1 | + 3.27·3-s + 0.246·5-s + 7.73·9-s − 11-s + 3.17·13-s + 0.808·15-s + 6.49·17-s − 4.32·19-s − 3.15·23-s − 4.93·25-s + 15.5·27-s + 6.48·29-s + 1.78·31-s − 3.27·33-s + 8.38·37-s + 10.3·39-s + 0.553·41-s − 5.69·43-s + 1.90·45-s + 10.2·47-s + 21.2·51-s − 10.1·53-s − 0.246·55-s − 14.1·57-s − 6.45·59-s + 3.38·61-s + 0.782·65-s + ⋯ |
| L(s) = 1 | + 1.89·3-s + 0.110·5-s + 2.57·9-s − 0.301·11-s + 0.879·13-s + 0.208·15-s + 1.57·17-s − 0.991·19-s − 0.658·23-s − 0.987·25-s + 2.98·27-s + 1.20·29-s + 0.319·31-s − 0.570·33-s + 1.37·37-s + 1.66·39-s + 0.0863·41-s − 0.867·43-s + 0.284·45-s + 1.50·47-s + 2.97·51-s − 1.39·53-s − 0.0332·55-s − 1.87·57-s − 0.840·59-s + 0.432·61-s + 0.0970·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.125823604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.125823604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 - 0.246T + 5T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 - 8.38T + 37T^{2} \) |
| 41 | \( 1 - 0.553T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 0.345T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 - 0.246T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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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
−7.988615515155739431430977995133, −7.42646991664678199049405063749, −6.45807559362778107803785291271, −5.85789419055774470588808032476, −4.68899671416889623888617235720, −4.02484149537937844146067732080, −3.38943278111677348762013125190, −2.71066124237815779102024370889, −1.94178768917431894947130623026, −1.07849551585873598538118195366,
1.07849551585873598538118195366, 1.94178768917431894947130623026, 2.71066124237815779102024370889, 3.38943278111677348762013125190, 4.02484149537937844146067732080, 4.68899671416889623888617235720, 5.85789419055774470588808032476, 6.45807559362778107803785291271, 7.42646991664678199049405063749, 7.988615515155739431430977995133
