| L(s) = 1 | + 2·2-s − 7·3-s − 4·4-s − 7·5-s − 14·6-s − 24·8-s + 22·9-s − 14·10-s − 5·11-s + 28·12-s + 14·13-s + 49·15-s − 16·16-s + 21·17-s + 44·18-s − 49·19-s + 28·20-s − 10·22-s − 159·23-s + 168·24-s − 76·25-s + 28·26-s + 35·27-s + 58·29-s + 98·30-s − 147·31-s + 160·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·3-s − 1/2·4-s − 0.626·5-s − 0.952·6-s − 1.06·8-s + 0.814·9-s − 0.442·10-s − 0.137·11-s + 0.673·12-s + 0.298·13-s + 0.843·15-s − 1/4·16-s + 0.299·17-s + 0.576·18-s − 0.591·19-s + 0.313·20-s − 0.0969·22-s − 1.44·23-s + 1.42·24-s − 0.607·25-s + 0.211·26-s + 0.249·27-s + 0.371·29-s + 0.596·30-s − 0.851·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 5 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 + 49 T + p^{3} T^{2} \) |
| 23 | \( 1 + 159 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 147 T + p^{3} T^{2} \) |
| 37 | \( 1 - 219 T + p^{3} T^{2} \) |
| 41 | \( 1 + 350 T + p^{3} T^{2} \) |
| 43 | \( 1 + 124 T + p^{3} T^{2} \) |
| 47 | \( 1 + 525 T + p^{3} T^{2} \) |
| 53 | \( 1 - 303 T + p^{3} T^{2} \) |
| 59 | \( 1 - 105 T + p^{3} T^{2} \) |
| 61 | \( 1 - 413 T + p^{3} T^{2} \) |
| 67 | \( 1 - 415 T + p^{3} T^{2} \) |
| 71 | \( 1 + 432 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1113 T + p^{3} T^{2} \) |
| 79 | \( 1 + 103 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1092 T + p^{3} T^{2} \) |
| 89 | \( 1 - 329 T + p^{3} T^{2} \) |
| 97 | \( 1 - 882 T + p^{3} T^{2} \) |
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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
−14.50681019136320017885788599755, −13.19248208451046490301064021177, −12.15471299621404812295175230070, −11.41404420563535358004270734252, −10.01022925353519610055249637935, −8.250433162401923681362381644552, −6.37184258556772176707180632096, −5.25402214220430272371979073517, −3.94043229564655617480298696766, 0,
3.94043229564655617480298696766, 5.25402214220430272371979073517, 6.37184258556772176707180632096, 8.250433162401923681362381644552, 10.01022925353519610055249637935, 11.41404420563535358004270734252, 12.15471299621404812295175230070, 13.19248208451046490301064021177, 14.50681019136320017885788599755
