| L(s) = 1 | − 0.321·2-s + 2.86·3-s − 1.89·4-s − 0.920·6-s − 0.829·7-s + 1.25·8-s + 5.18·9-s − 1.61·11-s − 5.42·12-s − 3.76·13-s + 0.266·14-s + 3.38·16-s − 2.25·17-s − 1.66·18-s + 5.21·19-s − 2.37·21-s + 0.520·22-s + 0.268·23-s + 3.58·24-s + 1.21·26-s + 6.24·27-s + 1.57·28-s − 6.37·29-s + 0.201·31-s − 3.59·32-s − 4.62·33-s + 0.725·34-s + ⋯ |
| L(s) = 1 | − 0.227·2-s + 1.65·3-s − 0.948·4-s − 0.375·6-s − 0.313·7-s + 0.443·8-s + 1.72·9-s − 0.487·11-s − 1.56·12-s − 1.04·13-s + 0.0713·14-s + 0.847·16-s − 0.547·17-s − 0.393·18-s + 1.19·19-s − 0.517·21-s + 0.110·22-s + 0.0559·23-s + 0.732·24-s + 0.237·26-s + 1.20·27-s + 0.297·28-s − 1.18·29-s + 0.0362·31-s − 0.636·32-s − 0.805·33-s + 0.124·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 0.321T + 2T^{2} \) |
| 3 | \( 1 - 2.86T + 3T^{2} \) |
| 7 | \( 1 + 0.829T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 - 0.268T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 - 0.201T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 + 8.49T + 43T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 - 4.02T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 7.56T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
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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
−7.937881611201998708239402733640, −7.52901608853947672807117139903, −6.81483201885149868803583984628, −5.44589242574628705027085981739, −4.86549821908378579745718996839, −3.92132254390429415216292343705, −3.31103747073471707301262430071, −2.51732644497890736495429499493, −1.54472551587583614713312110723, 0,
1.54472551587583614713312110723, 2.51732644497890736495429499493, 3.31103747073471707301262430071, 3.92132254390429415216292343705, 4.86549821908378579745718996839, 5.44589242574628705027085981739, 6.81483201885149868803583984628, 7.52901608853947672807117139903, 7.937881611201998708239402733640
