| L(s) = 1 | − 7-s − 5·11-s − 2·13-s − 4·17-s + 6·19-s − 23-s + 29-s − 2·31-s − 5·37-s − 7·43-s − 2·47-s + 49-s − 6·53-s + 6·59-s + 3·67-s + 9·71-s − 12·73-s + 5·77-s − 3·79-s − 6·83-s − 6·89-s + 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.50·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s − 0.208·23-s + 0.185·29-s − 0.359·31-s − 0.821·37-s − 1.06·43-s − 0.291·47-s + 1/7·49-s − 0.824·53-s + 0.781·59-s + 0.366·67-s + 1.06·71-s − 1.40·73-s + 0.569·77-s − 0.337·79-s − 0.658·83-s − 0.635·89-s + 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4311921830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4311921830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.58963574175093, −13.33534988878514, −12.81385339087254, −12.31661437758390, −11.86337377352068, −11.16920536885358, −10.91192166209107, −10.19207455084074, −9.789453926211928, −9.519139022366899, −8.546002030795949, −8.435222103873326, −7.652177522970039, −7.189826046835371, −6.854301601068877, −6.029030309703460, −5.557553612549462, −4.957797836677733, −4.674902481443329, −3.739977564951538, −3.205965927401944, −2.633267346731467, −2.098928417701981, −1.264416765244523, −0.2049357209277609,
0.2049357209277609, 1.264416765244523, 2.098928417701981, 2.633267346731467, 3.205965927401944, 3.739977564951538, 4.674902481443329, 4.957797836677733, 5.557553612549462, 6.029030309703460, 6.854301601068877, 7.189826046835371, 7.652177522970039, 8.435222103873326, 8.546002030795949, 9.519139022366899, 9.789453926211928, 10.19207455084074, 10.91192166209107, 11.16920536885358, 11.86337377352068, 12.31661437758390, 12.81385339087254, 13.33534988878514, 13.58963574175093
