L(s) = 1 | + 3-s + 9-s − 2·13-s + 6·17-s + 4·19-s + 4·23-s + 27-s + 6·29-s + 8·31-s + 10·37-s − 2·39-s + 10·41-s − 12·43-s − 8·47-s + 6·51-s − 6·53-s + 4·57-s − 4·59-s + 10·61-s − 12·67-s + 4·69-s + 4·71-s + 2·73-s + 8·79-s + 81-s + 4·83-s + 6·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.64·37-s − 0.320·39-s + 1.56·41-s − 1.82·43-s − 1.16·47-s + 0.840·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s − 1.46·67-s + 0.481·69-s + 0.474·71-s + 0.234·73-s + 0.900·79-s + 1/9·81-s + 0.439·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.675576914\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.675576914\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.02347830736929, −14.60326821802695, −14.21238330784629, −13.59209760325311, −13.13319714884180, −12.48745511186442, −12.00695525292706, −11.52627279314355, −10.85358248506356, −10.06436680579987, −9.775331347208411, −9.353140773919024, −8.514842189242590, −7.947513658931576, −7.680518854097384, −6.890527222872790, −6.338087410124866, −5.607247597369898, −4.859260745567523, −4.506854807833460, −3.459987052697289, −3.026339168557931, −2.460344480659377, −1.356261749549409, −0.7921002605257131,
0.7921002605257131, 1.356261749549409, 2.460344480659377, 3.026339168557931, 3.459987052697289, 4.506854807833460, 4.859260745567523, 5.607247597369898, 6.338087410124866, 6.890527222872790, 7.680518854097384, 7.947513658931576, 8.514842189242590, 9.353140773919024, 9.775331347208411, 10.06436680579987, 10.85358248506356, 11.52627279314355, 12.00695525292706, 12.48745511186442, 13.13319714884180, 13.59209760325311, 14.21238330784629, 14.60326821802695, 15.02347830736929