| L(s) = 1 | + 3·3-s + 5·5-s − 12·7-s + 9·9-s − 24·11-s + 38·13-s + 15·15-s − 6·17-s + 104·19-s − 36·21-s + 100·23-s + 25·25-s + 27·27-s + 230·29-s − 56·31-s − 72·33-s − 60·35-s + 190·37-s + 114·39-s + 202·41-s − 148·43-s + 45·45-s + 124·47-s − 199·49-s − 18·51-s + 206·53-s − 120·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.647·7-s + 1/3·9-s − 0.657·11-s + 0.810·13-s + 0.258·15-s − 0.0856·17-s + 1.25·19-s − 0.374·21-s + 0.906·23-s + 1/5·25-s + 0.192·27-s + 1.47·29-s − 0.324·31-s − 0.379·33-s − 0.289·35-s + 0.844·37-s + 0.468·39-s + 0.769·41-s − 0.524·43-s + 0.149·45-s + 0.384·47-s − 0.580·49-s − 0.0494·51-s + 0.533·53-s − 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.558464332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558464332\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 - 100 T + p^{3} T^{2} \) |
| 29 | \( 1 - 230 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 - 190 T + p^{3} T^{2} \) |
| 41 | \( 1 - 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 124 T + p^{3} T^{2} \) |
| 53 | \( 1 - 206 T + p^{3} T^{2} \) |
| 59 | \( 1 + 128 T + p^{3} T^{2} \) |
| 61 | \( 1 - 190 T + p^{3} T^{2} \) |
| 67 | \( 1 + 204 T + p^{3} T^{2} \) |
| 71 | \( 1 + 440 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 79 | \( 1 - 816 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1412 T + p^{3} T^{2} \) |
| 89 | \( 1 + 214 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1202 T + p^{3} T^{2} \) |
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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
−10.41622749054050663593390516684, −9.641939411294639579404550347131, −8.876707176376890709899324776573, −7.921304657796372278118400262727, −6.92189992030718392676078820046, −5.95122812575480884998599674005, −4.85378246152528777255751462926, −3.45768690804603567561083791554, −2.59956268418527990492689575188, −1.02473300525567783514756788390,
1.02473300525567783514756788390, 2.59956268418527990492689575188, 3.45768690804603567561083791554, 4.85378246152528777255751462926, 5.95122812575480884998599674005, 6.92189992030718392676078820046, 7.921304657796372278118400262727, 8.876707176376890709899324776573, 9.641939411294639579404550347131, 10.41622749054050663593390516684
