| L(s) = 1 | − 9.22·3-s − 9.91·5-s − 18.1·7-s + 58.0·9-s − 30.3·11-s + 38.5·13-s + 91.4·15-s + 59.0·17-s − 50.7·19-s + 167.·21-s + 155.·23-s − 26.6·25-s − 286.·27-s − 29·29-s − 241.·31-s + 279.·33-s + 179.·35-s − 371.·37-s − 355.·39-s + 61.4·41-s − 369.·43-s − 575.·45-s − 281.·47-s − 13.7·49-s − 545.·51-s + 374.·53-s + 300.·55-s + ⋯ |
| L(s) = 1 | − 1.77·3-s − 0.887·5-s − 0.979·7-s + 2.15·9-s − 0.831·11-s + 0.823·13-s + 1.57·15-s + 0.843·17-s − 0.612·19-s + 1.73·21-s + 1.41·23-s − 0.213·25-s − 2.04·27-s − 0.185·29-s − 1.39·31-s + 1.47·33-s + 0.869·35-s − 1.65·37-s − 1.46·39-s + 0.234·41-s − 1.31·43-s − 1.90·45-s − 0.874·47-s − 0.0400·49-s − 1.49·51-s + 0.969·53-s + 0.737·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1571094991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1571094991\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 9.22T + 27T^{2} \) |
| 5 | \( 1 + 9.91T + 125T^{2} \) |
| 7 | \( 1 + 18.1T + 343T^{2} \) |
| 11 | \( 1 + 30.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 371.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 61.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 374.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 129.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 463.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 970.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 241.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 565.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 424.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 161.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−8.934368969031330432730862355010, −7.88848285671483418885561656616, −7.05595908742856198290422872987, −6.50968436623415758718421361485, −5.58481175058152071930614501315, −5.07848579757850947490342620260, −3.95924242505833978156138775434, −3.22280750525519146151861867644, −1.43431315216554725002984207038, −0.21416710628756728758618672224,
0.21416710628756728758618672224, 1.43431315216554725002984207038, 3.22280750525519146151861867644, 3.95924242505833978156138775434, 5.07848579757850947490342620260, 5.58481175058152071930614501315, 6.50968436623415758718421361485, 7.05595908742856198290422872987, 7.88848285671483418885561656616, 8.934368969031330432730862355010
