L(s) = 1 | + 0.566·2-s + 0.484·3-s − 1.67·4-s + 0.274·6-s + 0.0113·7-s − 2.08·8-s − 2.76·9-s − 0.725·11-s − 0.814·12-s + 3.41·13-s + 0.00644·14-s + 2.17·16-s − 3.34·17-s − 1.56·18-s + 2.32·19-s + 0.00552·21-s − 0.410·22-s + 4.69·23-s − 1.01·24-s + 1.93·26-s − 2.79·27-s − 0.0191·28-s + 5.86·29-s − 0.631·31-s + 5.40·32-s − 0.351·33-s − 1.89·34-s + ⋯ |
L(s) = 1 | + 0.400·2-s + 0.279·3-s − 0.839·4-s + 0.112·6-s + 0.00430·7-s − 0.736·8-s − 0.921·9-s − 0.218·11-s − 0.235·12-s + 0.947·13-s + 0.00172·14-s + 0.544·16-s − 0.811·17-s − 0.369·18-s + 0.534·19-s + 0.00120·21-s − 0.0876·22-s + 0.978·23-s − 0.206·24-s + 0.379·26-s − 0.537·27-s − 0.00361·28-s + 1.08·29-s − 0.113·31-s + 0.954·32-s − 0.0612·33-s − 0.324·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 0.566T + 2T^{2} \) |
| 3 | \( 1 - 0.484T + 3T^{2} \) |
| 7 | \( 1 - 0.0113T + 7T^{2} \) |
| 11 | \( 1 + 0.725T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 + 0.631T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 + 0.320T + 41T^{2} \) |
| 43 | \( 1 - 3.06T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.842T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 + 8.96T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 5.27T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 - 0.282T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.079877996316306359687512223025, −6.85016687459322593289537007456, −6.24685604857773556853405003573, −5.38940127357167247118368162840, −4.92077094598013118113479235653, −3.99885800307496475662740815187, −3.27427176720331273609077707619, −2.64085998901791453130112704792, −1.25005305536879817149197231623, 0,
1.25005305536879817149197231623, 2.64085998901791453130112704792, 3.27427176720331273609077707619, 3.99885800307496475662740815187, 4.92077094598013118113479235653, 5.38940127357167247118368162840, 6.24685604857773556853405003573, 6.85016687459322593289537007456, 8.079877996316306359687512223025