| L(s) = 1 | − 2-s − 1.24·3-s + 4-s − 1.11·5-s + 1.24·6-s + 1.91·7-s − 8-s − 1.45·9-s + 1.11·10-s + 3.49·11-s − 1.24·12-s + 0.165·13-s − 1.91·14-s + 1.37·15-s + 16-s + 1.61·17-s + 1.45·18-s − 0.652·19-s − 1.11·20-s − 2.37·21-s − 3.49·22-s − 8.69·23-s + 1.24·24-s − 3.76·25-s − 0.165·26-s + 5.53·27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.716·3-s + 0.5·4-s − 0.497·5-s + 0.506·6-s + 0.724·7-s − 0.353·8-s − 0.486·9-s + 0.351·10-s + 1.05·11-s − 0.358·12-s + 0.0459·13-s − 0.512·14-s + 0.356·15-s + 0.250·16-s + 0.390·17-s + 0.344·18-s − 0.149·19-s − 0.248·20-s − 0.519·21-s − 0.745·22-s − 1.81·23-s + 0.253·24-s − 0.752·25-s − 0.0324·26-s + 1.06·27-s + 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 - 0.165T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 0.652T + 19T^{2} \) |
| 23 | \( 1 + 8.69T + 23T^{2} \) |
| 29 | \( 1 - 4.71T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 9.20T + 41T^{2} \) |
| 43 | \( 1 + 4.01T + 43T^{2} \) |
| 47 | \( 1 + 0.404T + 47T^{2} \) |
| 53 | \( 1 - 0.182T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 - 2.84T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.04T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 0.388T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.176924813249780925989516348701, −7.50114069058396607861474643598, −6.52334606798034273613623446271, −6.06362707824890938439942257316, −5.18453990200073141496877438634, −4.27235663587396973108348806760, −3.46098769378154638317750811146, −2.19508895695745019780172467371, −1.19420528843612210933365844526, 0,
1.19420528843612210933365844526, 2.19508895695745019780172467371, 3.46098769378154638317750811146, 4.27235663587396973108348806760, 5.18453990200073141496877438634, 6.06362707824890938439942257316, 6.52334606798034273613623446271, 7.50114069058396607861474643598, 8.176924813249780925989516348701
