| L(s) = 1 | − 1.00·2-s + 0.387·3-s − 6.99·4-s − 15.3·5-s − 0.388·6-s + 15.0·8-s − 26.8·9-s + 15.3·10-s + 67.1·11-s − 2.70·12-s − 17.1·13-s − 5.93·15-s + 40.8·16-s − 79.7·17-s + 26.9·18-s − 124.·19-s + 107.·20-s − 67.3·22-s − 62.3·23-s + 5.81·24-s + 110.·25-s + 17.1·26-s − 20.8·27-s + 213.·29-s + 5.95·30-s + 31.0·31-s − 161.·32-s + ⋯ |
| L(s) = 1 | − 0.354·2-s + 0.0744·3-s − 0.874·4-s − 1.37·5-s − 0.0264·6-s + 0.664·8-s − 0.994·9-s + 0.486·10-s + 1.84·11-s − 0.0651·12-s − 0.365·13-s − 0.102·15-s + 0.638·16-s − 1.13·17-s + 0.352·18-s − 1.50·19-s + 1.19·20-s − 0.652·22-s − 0.564·23-s + 0.0494·24-s + 0.881·25-s + 0.129·26-s − 0.148·27-s + 1.36·29-s + 0.0362·30-s + 0.179·31-s − 0.890·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 + 1.00T + 8T^{2} \) |
| 3 | \( 1 - 0.387T + 27T^{2} \) |
| 5 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 - 67.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 17.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 146.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 249.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 278.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 442.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 569.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 249.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 622.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 481.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.27T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.73e3T + 9.12e5T^{2} \) |
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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
−8.497709535555144259896282609291, −7.969978981918662823490884674134, −6.84810868554045490967288792288, −6.23723443989774149493775499584, −4.89421162899761193923304714895, −4.10729976494746671796751232367, −3.77829378347467046756704267900, −2.35910610000277521207472101917, −0.853051297098100908461344159292, 0,
0.853051297098100908461344159292, 2.35910610000277521207472101917, 3.77829378347467046756704267900, 4.10729976494746671796751232367, 4.89421162899761193923304714895, 6.23723443989774149493775499584, 6.84810868554045490967288792288, 7.969978981918662823490884674134, 8.497709535555144259896282609291
