L(s) = 1 | − 2-s − 2.09·3-s + 4-s + 1.88·5-s + 2.09·6-s − 4.81·7-s − 8-s + 1.40·9-s − 1.88·10-s − 2.76·11-s − 2.09·12-s + 1.57·13-s + 4.81·14-s − 3.95·15-s + 16-s − 7.15·17-s − 1.40·18-s − 19-s + 1.88·20-s + 10.1·21-s + 2.76·22-s − 0.301·23-s + 2.09·24-s − 1.45·25-s − 1.57·26-s + 3.34·27-s − 4.81·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.21·3-s + 0.5·4-s + 0.842·5-s + 0.857·6-s − 1.82·7-s − 0.353·8-s + 0.469·9-s − 0.595·10-s − 0.833·11-s − 0.606·12-s + 0.435·13-s + 1.28·14-s − 1.02·15-s + 0.250·16-s − 1.73·17-s − 0.331·18-s − 0.229·19-s + 0.421·20-s + 2.20·21-s + 0.589·22-s − 0.0628·23-s + 0.428·24-s − 0.290·25-s − 0.308·26-s + 0.643·27-s − 0.910·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 23 | \( 1 + 0.301T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 + 0.603T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 47 | \( 1 + 0.389T + 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 + 0.714T + 83T^{2} \) |
| 89 | \( 1 + 4.35T + 89T^{2} \) |
| 97 | \( 1 - 6.13T + 97T^{2} \) |
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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
−7.16643331655808666685551078660, −6.68685345815655121615072302654, −6.05108697811272429413233033471, −5.87168570780818040865339478245, −4.87578172104104883328713279231, −3.91614313715481292088233998410, −2.76975405987800875151444088944, −2.27570813605702175372663823597, −0.812050586587881376216587940266, 0,
0.812050586587881376216587940266, 2.27570813605702175372663823597, 2.76975405987800875151444088944, 3.91614313715481292088233998410, 4.87578172104104883328713279231, 5.87168570780818040865339478245, 6.05108697811272429413233033471, 6.68685345815655121615072302654, 7.16643331655808666685551078660