| L(s) = 1 | − 0.715·3-s − 5-s − 3.76·7-s − 2.48·9-s + 1.96·11-s + 0.715·15-s − 5.65·17-s − 0.336·19-s + 2.69·21-s + 1.38·23-s + 25-s + 3.92·27-s − 1.05·29-s − 9.97·31-s − 1.40·33-s + 3.76·35-s + 0.428·37-s − 3.99·41-s + 1.41·43-s + 2.48·45-s − 10.5·47-s + 7.16·49-s + 4.04·51-s − 0.428·53-s − 1.96·55-s + 0.240·57-s − 8.47·59-s + ⋯ |
| L(s) = 1 | − 0.413·3-s − 0.447·5-s − 1.42·7-s − 0.829·9-s + 0.592·11-s + 0.184·15-s − 1.37·17-s − 0.0771·19-s + 0.587·21-s + 0.288·23-s + 0.200·25-s + 0.755·27-s − 0.195·29-s − 1.79·31-s − 0.244·33-s + 0.636·35-s + 0.0703·37-s − 0.624·41-s + 0.216·43-s + 0.370·45-s − 1.53·47-s + 1.02·49-s + 0.567·51-s − 0.0589·53-s − 0.264·55-s + 0.0318·57-s − 1.10·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5542889283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5542889283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.715T + 3T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 - 1.96T + 11T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 0.336T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 31 | \( 1 + 9.97T + 31T^{2} \) |
| 37 | \( 1 - 0.428T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.428T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 8.25T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 9.33T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 9.45T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.828948992696760973109430131238, −7.86315849914171744572151267496, −6.75954419522866031837062178268, −6.58839183351059406961277917159, −5.68718650582565126351953350350, −4.82881586896301268696554986885, −3.77197079209205220932872280506, −3.21510511083921273145182358251, −2.10618544760363159502973440591, −0.42499448230794507774849499281,
0.42499448230794507774849499281, 2.10618544760363159502973440591, 3.21510511083921273145182358251, 3.77197079209205220932872280506, 4.82881586896301268696554986885, 5.68718650582565126351953350350, 6.58839183351059406961277917159, 6.75954419522866031837062178268, 7.86315849914171744572151267496, 8.828948992696760973109430131238
