| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 6·13-s + 3·17-s + 5·19-s + 3·21-s − 2·23-s − 9·27-s − 5·29-s − 5·31-s − 37-s − 18·39-s + 2·41-s + 12·43-s − 2·47-s − 6·49-s − 9·51-s − 13·53-s − 15·57-s + 2·59-s + 61-s − 6·63-s − 16·67-s + 6·69-s − 15·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 1.66·13-s + 0.727·17-s + 1.14·19-s + 0.654·21-s − 0.417·23-s − 1.73·27-s − 0.928·29-s − 0.898·31-s − 0.164·37-s − 2.88·39-s + 0.312·41-s + 1.82·43-s − 0.291·47-s − 6/7·49-s − 1.26·51-s − 1.78·53-s − 1.98·57-s + 0.260·59-s + 0.128·61-s − 0.755·63-s − 1.95·67-s + 0.722·69-s − 1.78·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.12500541717306, −12.81336616760012, −12.29769470002843, −11.90700103652445, −11.28314904384472, −11.14124791868122, −10.67667162359194, −10.15182320672775, −9.655484318832099, −9.204061372660427, −8.688272046900980, −7.775720077803648, −7.591489876953817, −6.974948107330869, −6.351349323996039, −5.938408719074317, −5.726046470371596, −5.198046051242359, −4.544988602698891, −3.995770385171763, −3.457651678978604, −2.954628904952250, −1.761016405066840, −1.367848591949295, −0.7416468744116336, 0,
0.7416468744116336, 1.367848591949295, 1.761016405066840, 2.954628904952250, 3.457651678978604, 3.995770385171763, 4.544988602698891, 5.198046051242359, 5.726046470371596, 5.938408719074317, 6.351349323996039, 6.974948107330869, 7.591489876953817, 7.775720077803648, 8.688272046900980, 9.204061372660427, 9.655484318832099, 10.15182320672775, 10.67667162359194, 11.14124791868122, 11.28314904384472, 11.90700103652445, 12.29769470002843, 12.81336616760012, 13.12500541717306
