| L(s) = 1 | + 2-s − 3-s + 4-s + 1.37·5-s − 6-s + 7-s + 8-s + 9-s + 1.37·10-s − 4·11-s − 12-s − 4.37·13-s + 14-s − 1.37·15-s + 16-s + 6.37·17-s + 18-s + 19-s + 1.37·20-s − 21-s − 4·22-s − 1.37·23-s − 24-s − 3.11·25-s − 4.37·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.613·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.433·10-s − 1.20·11-s − 0.288·12-s − 1.21·13-s + 0.267·14-s − 0.354·15-s + 0.250·16-s + 1.54·17-s + 0.235·18-s + 0.229·19-s + 0.306·20-s − 0.218·21-s − 0.852·22-s − 0.286·23-s − 0.204·24-s − 0.623·25-s − 0.857·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 9.74T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4.62T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 3.37T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 9.62T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.51715829377384139900637318540, −7.12091579481669260893644358788, −5.85975714003520723249340564842, −5.54477529737974675656487527398, −5.11670279887432892457223279075, −4.17968577997510991180517746350, −3.24693868720888560363783390703, −2.32439396016518463395461094587, −1.56326723276163181775220078671, 0,
1.56326723276163181775220078671, 2.32439396016518463395461094587, 3.24693868720888560363783390703, 4.17968577997510991180517746350, 5.11670279887432892457223279075, 5.54477529737974675656487527398, 5.85975714003520723249340564842, 7.12091579481669260893644358788, 7.51715829377384139900637318540
