| L(s) = 1 | − 2.30·3-s + 1.30·5-s − 7-s + 2.30·9-s − 0.605·13-s − 3·15-s + 17-s + 0.605·19-s + 2.30·21-s − 3.30·25-s + 1.60·27-s − 0.697·31-s − 1.30·35-s + 4.60·37-s + 1.39·39-s − 6.90·41-s − 3.69·43-s + 3.00·45-s + 2.60·47-s + 49-s − 2.30·51-s − 7.30·53-s − 1.39·57-s + 5.21·59-s + 2.90·61-s − 2.30·63-s − 0.788·65-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.582·5-s − 0.377·7-s + 0.767·9-s − 0.167·13-s − 0.774·15-s + 0.242·17-s + 0.138·19-s + 0.502·21-s − 0.660·25-s + 0.308·27-s − 0.125·31-s − 0.220·35-s + 0.757·37-s + 0.223·39-s − 1.07·41-s − 0.563·43-s + 0.447·45-s + 0.380·47-s + 0.142·49-s − 0.322·51-s − 1.00·53-s − 0.184·57-s + 0.678·59-s + 0.372·61-s − 0.290·63-s − 0.0978·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 19 | \( 1 - 0.605T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 0.697T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 + 3.69T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 9.39T + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
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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
−8.961764246845065996988789796897, −7.952243132294192252932079922511, −6.96611596022818501071849884563, −6.31340954288137810848601581639, −5.63279453660335957150858556547, −5.02650048285453017710625106590, −3.96839751832559132028982404526, −2.73445676618642352096055896486, −1.40604163485907181033254461841, 0,
1.40604163485907181033254461841, 2.73445676618642352096055896486, 3.96839751832559132028982404526, 5.02650048285453017710625106590, 5.63279453660335957150858556547, 6.31340954288137810848601581639, 6.96611596022818501071849884563, 7.952243132294192252932079922511, 8.961764246845065996988789796897
