| L(s) = 1 | + 2·3-s − 7-s + 9-s − 2·11-s + 4·13-s + 2·17-s − 2·21-s − 23-s − 4·27-s + 2·29-s − 4·33-s − 4·37-s + 8·39-s + 6·41-s + 2·43-s − 4·47-s + 49-s + 4·51-s − 2·59-s + 10·61-s − 63-s + 2·67-s − 2·69-s − 8·71-s + 6·73-s + 2·77-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.485·17-s − 0.436·21-s − 0.208·23-s − 0.769·27-s + 0.371·29-s − 0.696·33-s − 0.657·37-s + 1.28·39-s + 0.937·41-s + 0.304·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s − 0.260·59-s + 1.28·61-s − 0.125·63-s + 0.244·67-s − 0.240·69-s − 0.949·71-s + 0.702·73-s + 0.227·77-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.27859930160333, −12.63043599838498, −12.32884868202021, −11.59369699376562, −11.24459146801374, −10.62675414224076, −10.27956754016599, −9.689042716503176, −9.324125567327276, −8.795923555660697, −8.370699174918116, −8.067048647252510, −7.509093819894380, −7.084898645728808, −6.350739344491235, −6.013226168534320, −5.407559955844421, −4.908885286217876, −4.088479602246751, −3.730188064496023, −3.250761467108190, −2.715538370563216, −2.263394324503651, −1.551822420270607, −0.8949687948514274, 0,
0.8949687948514274, 1.551822420270607, 2.263394324503651, 2.715538370563216, 3.250761467108190, 3.730188064496023, 4.088479602246751, 4.908885286217876, 5.407559955844421, 6.013226168534320, 6.350739344491235, 7.084898645728808, 7.509093819894380, 8.067048647252510, 8.370699174918116, 8.795923555660697, 9.324125567327276, 9.689042716503176, 10.27956754016599, 10.62675414224076, 11.24459146801374, 11.59369699376562, 12.32884868202021, 12.63043599838498, 13.27859930160333
