| 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
| \(L(1)\) |
\(\approx\) |
\(2.770311683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.770311683\) |
| \(L(\frac{3}{2})\) |
|
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
−9.074628956005803176291849209079, −7.76756939281618445419042689503, −7.22435932629250266803230764594, −6.32568066442545046843171801220, −5.40935788437607906839024954513, −5.12746318851386392790600866198, −4.39070869761718807480435939426, −2.72465994047367968737909268689, −1.90322466721907683805865987541, −1.16617178415852272826604714643,
1.16617178415852272826604714643, 1.90322466721907683805865987541, 2.72465994047367968737909268689, 4.39070869761718807480435939426, 5.12746318851386392790600866198, 5.40935788437607906839024954513, 6.32568066442545046843171801220, 7.22435932629250266803230764594, 7.76756939281618445419042689503, 9.074628956005803176291849209079
