L(s) = 1 | + 3-s − 4.02·5-s + 4.61·7-s + 9-s − 1.53·11-s + 5.57·13-s − 4.02·15-s − 1.29·17-s − 4.35·19-s + 4.61·21-s + 4·23-s + 11.2·25-s + 27-s + 1.86·29-s − 2.77·31-s − 1.53·33-s − 18.5·35-s − 3.97·37-s + 5.57·39-s − 3.03·41-s − 3.03·43-s − 4.02·45-s + 9.65·47-s + 14.2·49-s − 1.29·51-s + 8.58·53-s + 6.16·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.80·5-s + 1.74·7-s + 0.333·9-s − 0.461·11-s + 1.54·13-s − 1.03·15-s − 0.314·17-s − 1.00·19-s + 1.00·21-s + 0.834·23-s + 2.24·25-s + 0.192·27-s + 0.345·29-s − 0.498·31-s − 0.266·33-s − 3.14·35-s − 0.653·37-s + 0.893·39-s − 0.473·41-s − 0.462·43-s − 0.600·45-s + 1.40·47-s + 2.04·49-s − 0.181·51-s + 1.17·53-s + 0.831·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.066749267\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066749267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 + 3.03T + 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 + 5.73T + 59T^{2} \) |
| 61 | \( 1 - 7.64T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 6.88T + 71T^{2} \) |
| 73 | \( 1 + 3.50T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 + 4.12T + 83T^{2} \) |
| 89 | \( 1 - 7.98T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−8.602899645145906112070604023252, −8.094021781101200719769904461755, −7.45312672797873671083407921241, −6.75440912100718208903190757390, −5.43691734236198459298487517096, −4.56640407360088532121020784018, −4.04972207316457399489160784624, −3.27772814478544598419385901120, −2.03408503178541237218030789552, −0.885385341706773391849892376532,
0.885385341706773391849892376532, 2.03408503178541237218030789552, 3.27772814478544598419385901120, 4.04972207316457399489160784624, 4.56640407360088532121020784018, 5.43691734236198459298487517096, 6.75440912100718208903190757390, 7.45312672797873671083407921241, 8.094021781101200719769904461755, 8.602899645145906112070604023252