| L(s) = 1 | + 3·2-s + 4-s − 18·5-s − 21·8-s − 54·10-s + 36·11-s + 34·13-s − 71·16-s + 42·17-s + 124·19-s − 18·20-s + 108·22-s + 199·25-s + 102·26-s − 102·29-s + 160·31-s − 45·32-s + 126·34-s + 398·37-s + 372·38-s + 378·40-s − 318·41-s − 268·43-s + 36·44-s + 240·47-s + 597·50-s + 34·52-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 1.60·5-s − 0.928·8-s − 1.70·10-s + 0.986·11-s + 0.725·13-s − 1.10·16-s + 0.599·17-s + 1.49·19-s − 0.201·20-s + 1.04·22-s + 1.59·25-s + 0.769·26-s − 0.653·29-s + 0.926·31-s − 0.248·32-s + 0.635·34-s + 1.76·37-s + 1.58·38-s + 1.49·40-s − 1.21·41-s − 0.950·43-s + 0.123·44-s + 0.744·47-s + 1.68·50-s + 0.0906·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.244917358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244917358\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 318 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 - 498 T + p^{3} T^{2} \) |
| 59 | \( 1 + 132 T + p^{3} T^{2} \) |
| 61 | \( 1 + 398 T + p^{3} T^{2} \) |
| 67 | \( 1 - 92 T + p^{3} T^{2} \) |
| 71 | \( 1 - 720 T + p^{3} T^{2} \) |
| 73 | \( 1 - 502 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 83 | \( 1 + 204 T + p^{3} T^{2} \) |
| 89 | \( 1 - 354 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
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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
−11.26014131280576601338953214315, −9.764820151949806553705547112161, −8.767951993539021817750960840129, −7.889662222107988959834746006944, −6.88834055361016540511882653549, −5.78649557724850355316900052586, −4.65098036948284657545615792206, −3.79915332853922999824620412539, −3.19662142771194618453947502696, −0.841888082827454654369955675858,
0.841888082827454654369955675858, 3.19662142771194618453947502696, 3.79915332853922999824620412539, 4.65098036948284657545615792206, 5.78649557724850355316900052586, 6.88834055361016540511882653549, 7.889662222107988959834746006944, 8.767951993539021817750960840129, 9.764820151949806553705547112161, 11.26014131280576601338953214315
