| L(s) = 1 | − 2-s − 4-s − 4·5-s + 3·8-s + 4·10-s − 4·11-s − 2·13-s − 16-s + 2·19-s + 4·20-s + 4·22-s + 11·25-s + 2·26-s − 2·31-s − 5·32-s + 2·37-s − 2·38-s − 12·40-s − 4·41-s + 8·43-s + 4·44-s + 8·47-s − 7·49-s − 11·50-s + 2·52-s + 12·53-s + 16·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s + 1.26·10-s − 1.20·11-s − 0.554·13-s − 1/4·16-s + 0.458·19-s + 0.894·20-s + 0.852·22-s + 11/5·25-s + 0.392·26-s − 0.359·31-s − 0.883·32-s + 0.328·37-s − 0.324·38-s − 1.89·40-s − 0.624·41-s + 1.21·43-s + 0.603·44-s + 1.16·47-s − 49-s − 1.55·50-s + 0.277·52-s + 1.64·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.002756197304046112434640551546, −7.45667088529545126081969799392, −7.04992744884305439240974253275, −5.57976303095066252860125123031, −4.87229192348647470997014645368, −4.20538337797815428482051051861, −3.46550211035964012411662135430, −2.44700107130059124462725972189, −0.862995531096279071165248521938, 0,
0.862995531096279071165248521938, 2.44700107130059124462725972189, 3.46550211035964012411662135430, 4.20538337797815428482051051861, 4.87229192348647470997014645368, 5.57976303095066252860125123031, 7.04992744884305439240974253275, 7.45667088529545126081969799392, 8.002756197304046112434640551546
