L(s) = 1 | − 1.94·2-s − 3-s + 1.77·4-s − 0.275·5-s + 1.94·6-s + 3.15·7-s + 0.435·8-s + 9-s + 0.536·10-s + 1.59·11-s − 1.77·12-s − 13-s − 6.12·14-s + 0.275·15-s − 4.39·16-s + 0.648·17-s − 1.94·18-s + 5.78·19-s − 0.489·20-s − 3.15·21-s − 3.09·22-s + 6.66·23-s − 0.435·24-s − 4.92·25-s + 1.94·26-s − 27-s + 5.59·28-s + ⋯ |
L(s) = 1 | − 1.37·2-s − 0.577·3-s + 0.887·4-s − 0.123·5-s + 0.793·6-s + 1.19·7-s + 0.154·8-s + 0.333·9-s + 0.169·10-s + 0.480·11-s − 0.512·12-s − 0.277·13-s − 1.63·14-s + 0.0712·15-s − 1.09·16-s + 0.157·17-s − 0.457·18-s + 1.32·19-s − 0.109·20-s − 0.688·21-s − 0.660·22-s + 1.38·23-s − 0.0889·24-s − 0.984·25-s + 0.381·26-s − 0.192·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9603520086\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9603520086\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 + 0.275T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 17 | \( 1 - 0.648T + 17T^{2} \) |
| 19 | \( 1 - 5.78T + 19T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 5.10T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 - 5.08T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 - 2.66T + 67T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 - 4.40T + 73T^{2} \) |
| 79 | \( 1 - 1.43T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 + 3.51T + 97T^{2} \) |
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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
−8.388474711605332247163187057539, −7.75953537884886570692365019815, −7.32456739474559386471970096777, −6.49557696261154537019978126131, −5.43284258761581426569195263372, −4.81551781060656672390956596347, −3.96538374947006011421012637289, −2.57205283664856736017296132909, −1.44424632741177652731849475543, −0.810782774739208855444551401375,
0.810782774739208855444551401375, 1.44424632741177652731849475543, 2.57205283664856736017296132909, 3.96538374947006011421012637289, 4.81551781060656672390956596347, 5.43284258761581426569195263372, 6.49557696261154537019978126131, 7.32456739474559386471970096777, 7.75953537884886570692365019815, 8.388474711605332247163187057539