L(s) = 1 | + (−1 + i)3-s + i·9-s + (3 + 3i)11-s + (3 − 3i)13-s + 4·17-s + (−1 + i)19-s − 8i·23-s + (−4 − 4i)27-s + (−3 + 3i)29-s − 6·33-s + (3 + 3i)37-s + 6i·39-s + (3 + 3i)43-s − 2·47-s + 7·49-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.577i)3-s + 0.333i·9-s + (0.904 + 0.904i)11-s + (0.832 − 0.832i)13-s + 0.970·17-s + (−0.229 + 0.229i)19-s − 1.66i·23-s + (−0.769 − 0.769i)27-s + (−0.557 + 0.557i)29-s − 1.04·33-s + (0.493 + 0.493i)37-s + 0.960i·39-s + (0.457 + 0.457i)43-s − 0.291·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.495813695\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495813695\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + (-3 - 3i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + (1 - i)T - 19iT^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + (3 - 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9 - 9i)T + 59iT^{2} \) |
| 61 | \( 1 + (5 - 5i)T - 61iT^{2} \) |
| 67 | \( 1 + (3 - 3i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (9 - 9i)T - 83iT^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−9.702758765679975250072566101202, −8.812879179883277481024097487469, −7.998065423440694439071413149315, −7.13308311053660077800456212812, −6.14098217292904939195342161985, −5.49392581475311240114600842940, −4.51333780310260280007689941975, −3.85953434011924264483549117304, −2.56127311830190730740249320736, −1.13508494439198558991427077677,
0.78693737983005232923352362527, 1.76296657973499314889894800984, 3.44620474159706869408090868779, 3.98189993015752885571722268460, 5.51278277207434065107806723603, 5.97919179546714795891094934094, 6.78121466765948119591514506054, 7.49209862717289665177562414446, 8.542019788782969259470722357142, 9.218389907483313302817842802629