L(s) = 1 | − 1.14·2-s + 3-s − 0.694·4-s + 1.38·5-s − 1.14·6-s + 3.07·8-s + 9-s − 1.58·10-s − 3.33·11-s − 0.694·12-s + 0.235·13-s + 1.38·15-s − 2.12·16-s − 1.73·17-s − 1.14·18-s + 3.82·19-s − 0.962·20-s + 3.80·22-s + 0.816·23-s + 3.07·24-s − 3.07·25-s − 0.268·26-s + 27-s + 2.01·29-s − 1.58·30-s − 7.57·31-s − 3.72·32-s + ⋯ |
L(s) = 1 | − 0.808·2-s + 0.577·3-s − 0.347·4-s + 0.619·5-s − 0.466·6-s + 1.08·8-s + 0.333·9-s − 0.500·10-s − 1.00·11-s − 0.200·12-s + 0.0652·13-s + 0.357·15-s − 0.532·16-s − 0.421·17-s − 0.269·18-s + 0.876·19-s − 0.215·20-s + 0.812·22-s + 0.170·23-s + 0.628·24-s − 0.615·25-s − 0.0527·26-s + 0.192·27-s + 0.374·29-s − 0.289·30-s − 1.36·31-s − 0.658·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.375202719\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.375202719\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 - 0.235T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 0.816T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 + 7.57T + 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 4.90T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 2.06T + 89T^{2} \) |
| 97 | \( 1 - 5.96T + 97T^{2} \) |
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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
−8.234544075427686613802781316222, −7.43337298949846612303671182101, −7.10845092193639316765735266783, −5.80080446405909503018821522103, −5.31103844969532445530633797099, −4.41018105730443533461289761427, −3.59486421340088539138373930966, −2.53571538080223238862098672912, −1.81885261424978526673163822308, −0.68366170361068228039093143685,
0.68366170361068228039093143685, 1.81885261424978526673163822308, 2.53571538080223238862098672912, 3.59486421340088539138373930966, 4.41018105730443533461289761427, 5.31103844969532445530633797099, 5.80080446405909503018821522103, 7.10845092193639316765735266783, 7.43337298949846612303671182101, 8.234544075427686613802781316222