| L(s) = 1 | + 3·3-s + 10·7-s + 9·9-s + 14·11-s + 82·13-s − 18·17-s + 136·19-s + 30·21-s − 140·23-s + 27·27-s + 112·29-s − 72·31-s + 42·33-s − 26·37-s + 246·39-s − 446·41-s + 396·43-s − 144·47-s − 243·49-s − 54·51-s − 158·53-s + 408·57-s + 342·59-s + 314·61-s + 90·63-s − 152·67-s − 420·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.539·7-s + 1/3·9-s + 0.383·11-s + 1.74·13-s − 0.256·17-s + 1.64·19-s + 0.311·21-s − 1.26·23-s + 0.192·27-s + 0.717·29-s − 0.417·31-s + 0.221·33-s − 0.115·37-s + 1.01·39-s − 1.69·41-s + 1.40·43-s − 0.446·47-s − 0.708·49-s − 0.148·51-s − 0.409·53-s + 0.948·57-s + 0.754·59-s + 0.659·61-s + 0.179·63-s − 0.277·67-s − 0.732·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.411129019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411129019\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - 82 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 136 T + p^{3} T^{2} \) |
| 23 | \( 1 + 140 T + p^{3} T^{2} \) |
| 29 | \( 1 - 112 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 446 T + p^{3} T^{2} \) |
| 43 | \( 1 - 396 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 + 158 T + p^{3} T^{2} \) |
| 59 | \( 1 - 342 T + p^{3} T^{2} \) |
| 61 | \( 1 - 314 T + p^{3} T^{2} \) |
| 67 | \( 1 + 152 T + p^{3} T^{2} \) |
| 71 | \( 1 - 932 T + p^{3} T^{2} \) |
| 73 | \( 1 - 548 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 - 284 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1304 T + p^{3} T^{2} \) |
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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
−9.311117893407281311270998437073, −8.436753846923032004533268464973, −7.984515715160951759493984808184, −6.93014058246950242319153021130, −6.06305170082254992504223020024, −5.09042677076829796848510268530, −3.95552741811721020707830291853, −3.29597021837749466799334287989, −1.91914440033410951794585725344, −0.996717150605401770401358603409,
0.996717150605401770401358603409, 1.91914440033410951794585725344, 3.29597021837749466799334287989, 3.95552741811721020707830291853, 5.09042677076829796848510268530, 6.06305170082254992504223020024, 6.93014058246950242319153021130, 7.984515715160951759493984808184, 8.436753846923032004533268464973, 9.311117893407281311270998437073
