| L(s) = 1 | − 1.66·2-s − 3-s + 0.784·4-s + 5-s + 1.66·6-s − 7-s + 2.02·8-s + 9-s − 1.66·10-s − 5.31·11-s − 0.784·12-s − 4.52·13-s + 1.66·14-s − 15-s − 4.95·16-s − 6.34·17-s − 1.66·18-s − 0.456·19-s + 0.784·20-s + 21-s + 8.86·22-s + 2.14·23-s − 2.02·24-s + 25-s + 7.55·26-s − 27-s − 0.784·28-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 0.577·3-s + 0.392·4-s + 0.447·5-s + 0.681·6-s − 0.377·7-s + 0.716·8-s + 0.333·9-s − 0.527·10-s − 1.60·11-s − 0.226·12-s − 1.25·13-s + 0.446·14-s − 0.258·15-s − 1.23·16-s − 1.53·17-s − 0.393·18-s − 0.104·19-s + 0.175·20-s + 0.218·21-s + 1.88·22-s + 0.447·23-s − 0.413·24-s + 0.200·25-s + 1.48·26-s − 0.192·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 11 | \( 1 + 5.31T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 + 6.34T + 17T^{2} \) |
| 19 | \( 1 + 0.456T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 0.314T + 31T^{2} \) |
| 37 | \( 1 - 8.62T + 37T^{2} \) |
| 41 | \( 1 - 5.56T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 9.97T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 1.08T + 97T^{2} \) |
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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
−7.54735741487106331958591774203, −7.05477262203045582191354089469, −6.30137381380080579408956736149, −5.39499102485573154712116630784, −4.83203016785696107778129598676, −4.13783295297059027162749845250, −2.47088110586184801544470112150, −2.36728502157197297807927550314, −0.837753223054736450172110903166, 0,
0.837753223054736450172110903166, 2.36728502157197297807927550314, 2.47088110586184801544470112150, 4.13783295297059027162749845250, 4.83203016785696107778129598676, 5.39499102485573154712116630784, 6.30137381380080579408956736149, 7.05477262203045582191354089469, 7.54735741487106331958591774203
