| L(s) = 1 | − 8·7-s − 36·11-s + 10·13-s + 18·17-s − 100·19-s + 72·23-s + 234·29-s − 16·31-s + 226·37-s − 90·41-s − 452·43-s + 432·47-s − 279·49-s + 414·53-s + 684·59-s + 422·61-s − 332·67-s + 360·71-s − 26·73-s + 288·77-s + 512·79-s − 1.18e3·83-s + 630·89-s − 80·91-s + 1.05e3·97-s − 558·101-s − 8·103-s + ⋯ |
| L(s) = 1 | − 0.431·7-s − 0.986·11-s + 0.213·13-s + 0.256·17-s − 1.20·19-s + 0.652·23-s + 1.49·29-s − 0.0926·31-s + 1.00·37-s − 0.342·41-s − 1.60·43-s + 1.34·47-s − 0.813·49-s + 1.07·53-s + 1.50·59-s + 0.885·61-s − 0.605·67-s + 0.601·71-s − 0.0416·73-s + 0.426·77-s + 0.729·79-s − 1.57·83-s + 0.750·89-s − 0.0921·91-s + 1.10·97-s − 0.549·101-s − 0.00765·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.595261765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595261765\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 90 T + p^{3} T^{2} \) |
| 43 | \( 1 + 452 T + p^{3} T^{2} \) |
| 47 | \( 1 - 432 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 422 T + p^{3} T^{2} \) |
| 67 | \( 1 + 332 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 26 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1054 T + p^{3} T^{2} \) |
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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
−9.913944760707651460915990563691, −8.740576616468956049292679497809, −8.183917561380891923520411493255, −7.10284994686580048023996093454, −6.32329578586665332639393058923, −5.34028779150890163621722974439, −4.39631891205641441971053659601, −3.21680306025514240856597822597, −2.25004611792383487495098094015, −0.67272387326727346081879795116,
0.67272387326727346081879795116, 2.25004611792383487495098094015, 3.21680306025514240856597822597, 4.39631891205641441971053659601, 5.34028779150890163621722974439, 6.32329578586665332639393058923, 7.10284994686580048023996093454, 8.183917561380891923520411493255, 8.740576616468956049292679497809, 9.913944760707651460915990563691
