L(s) = 1 | − 2-s + 1.07·3-s + 4-s − 1.15·5-s − 1.07·6-s − 1.16·7-s − 8-s − 1.85·9-s + 1.15·10-s + 4.65·11-s + 1.07·12-s − 3.97·13-s + 1.16·14-s − 1.23·15-s + 16-s + 7.51·17-s + 1.85·18-s + 19-s − 1.15·20-s − 1.24·21-s − 4.65·22-s + 4.51·23-s − 1.07·24-s − 3.65·25-s + 3.97·26-s − 5.19·27-s − 1.16·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.618·3-s + 0.5·4-s − 0.517·5-s − 0.437·6-s − 0.438·7-s − 0.353·8-s − 0.617·9-s + 0.366·10-s + 1.40·11-s + 0.309·12-s − 1.10·13-s + 0.310·14-s − 0.320·15-s + 0.250·16-s + 1.82·17-s + 0.436·18-s + 0.229·19-s − 0.258·20-s − 0.271·21-s − 0.992·22-s + 0.941·23-s − 0.218·24-s − 0.731·25-s + 0.779·26-s − 1.00·27-s − 0.219·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.423398507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.423398507\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 - 4.65T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 23 | \( 1 - 4.51T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + 9.38T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 - 8.37T + 53T^{2} \) |
| 59 | \( 1 + 9.98T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 - 9.18T + 73T^{2} \) |
| 79 | \( 1 - 5.10T + 79T^{2} \) |
| 83 | \( 1 + 5.95T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−7.82008230086591631082356084178, −7.36228262878324775369062599598, −6.69040607931735343643296427310, −5.85456603992532501364333955663, −5.13253340214077511137099164683, −4.00073915845440712667531361769, −3.32692796871740882016589205920, −2.78419093690544414830678855080, −1.67187292177947434148197646446, −0.64468964187245518337834629602,
0.64468964187245518337834629602, 1.67187292177947434148197646446, 2.78419093690544414830678855080, 3.32692796871740882016589205920, 4.00073915845440712667531361769, 5.13253340214077511137099164683, 5.85456603992532501364333955663, 6.69040607931735343643296427310, 7.36228262878324775369062599598, 7.82008230086591631082356084178