| L(s) = 1 | + 1.72·2-s + 1.06·3-s + 0.984·4-s − 5-s + 1.83·6-s − 1.75·8-s − 1.87·9-s − 1.72·10-s + 0.439·11-s + 1.04·12-s + 4.67·13-s − 1.06·15-s − 4.99·16-s + 7.23·17-s − 3.23·18-s − 3.04·19-s − 0.984·20-s + 0.759·22-s − 8.70·23-s − 1.86·24-s + 25-s + 8.08·26-s − 5.17·27-s − 0.588·29-s − 1.83·30-s + 0.770·31-s − 5.13·32-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.612·3-s + 0.492·4-s − 0.447·5-s + 0.748·6-s − 0.620·8-s − 0.624·9-s − 0.546·10-s + 0.132·11-s + 0.301·12-s + 1.29·13-s − 0.274·15-s − 1.24·16-s + 1.75·17-s − 0.762·18-s − 0.699·19-s − 0.220·20-s + 0.161·22-s − 1.81·23-s − 0.380·24-s + 0.200·25-s + 1.58·26-s − 0.995·27-s − 0.109·29-s − 0.334·30-s + 0.138·31-s − 0.906·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 3 | \( 1 - 1.06T + 3T^{2} \) |
| 11 | \( 1 - 0.439T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 + 8.70T + 23T^{2} \) |
| 29 | \( 1 + 0.588T + 29T^{2} \) |
| 31 | \( 1 - 0.770T + 31T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 - 8.67T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 3.94T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 1.89T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 5.66T + 97T^{2} \) |
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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
−7.46657154756007906976365960232, −6.28651954382648721920207466333, −6.01135288396940605898454910160, −5.33497835990653595512577303626, −4.40434624228519059367304801001, −3.70329129020084326979662667403, −3.40114207765720256195954288741, −2.57769933161077173580384139346, −1.50023499678735959455351324713, 0,
1.50023499678735959455351324713, 2.57769933161077173580384139346, 3.40114207765720256195954288741, 3.70329129020084326979662667403, 4.40434624228519059367304801001, 5.33497835990653595512577303626, 6.01135288396940605898454910160, 6.28651954382648721920207466333, 7.46657154756007906976365960232
