| L(s) = 1 | + 2-s − 0.316·3-s + 4-s − 0.633·5-s − 0.316·6-s + 1.30·7-s + 8-s − 2.89·9-s − 0.633·10-s − 3.28·11-s − 0.316·12-s + 2.72·13-s + 1.30·14-s + 0.200·15-s + 16-s + 7.85·17-s − 2.89·18-s − 19-s − 0.633·20-s − 0.412·21-s − 3.28·22-s + 5.94·23-s − 0.316·24-s − 4.59·25-s + 2.72·26-s + 1.86·27-s + 1.30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.182·3-s + 0.5·4-s − 0.283·5-s − 0.129·6-s + 0.492·7-s + 0.353·8-s − 0.966·9-s − 0.200·10-s − 0.989·11-s − 0.0914·12-s + 0.754·13-s + 0.348·14-s + 0.0518·15-s + 0.250·16-s + 1.90·17-s − 0.683·18-s − 0.229·19-s − 0.141·20-s − 0.0899·21-s − 0.699·22-s + 1.23·23-s − 0.0646·24-s − 0.919·25-s + 0.533·26-s + 0.359·27-s + 0.246·28-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
| \(L(1)\) |
\(\approx\) |
\(2.844707928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.844707928\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.316T + 3T^{2} \) |
| 5 | \( 1 + 0.633T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 23 | \( 1 - 5.94T + 23T^{2} \) |
| 29 | \( 1 - 4.86T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.38T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 - 6.78T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 + 9.62T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.85735506163881680655266869720, −7.17650187058770157442920287815, −6.22263074204411366454123043316, −5.51223793577246496628435521347, −5.28002904511072303084321595984, −4.33985864780144206482869219291, −3.38415482242283295626470526114, −2.97535191013757664246028066295, −1.88480213409335156353032289792, −0.75814489572299258681022447534,
0.75814489572299258681022447534, 1.88480213409335156353032289792, 2.97535191013757664246028066295, 3.38415482242283295626470526114, 4.33985864780144206482869219291, 5.28002904511072303084321595984, 5.51223793577246496628435521347, 6.22263074204411366454123043316, 7.17650187058770157442920287815, 7.85735506163881680655266869720
