L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.03 + 0.928i)5-s + (−1.40 + 2.42i)7-s + 0.999·8-s + (−0.213 − 2.22i)10-s + (0.515 − 0.297i)11-s + (1.10 − 3.43i)13-s + 2.80·14-s + (−0.5 − 0.866i)16-s + (4.98 + 2.87i)17-s + (−6.59 − 3.80i)19-s + (−1.82 + 1.29i)20-s + (−0.515 − 0.297i)22-s + (4.02 − 2.32i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.909 + 0.415i)5-s + (−0.530 + 0.918i)7-s + 0.353·8-s + (−0.0673 − 0.703i)10-s + (0.155 − 0.0896i)11-s + (0.307 − 0.951i)13-s + 0.749·14-s + (−0.125 − 0.216i)16-s + (1.20 + 0.697i)17-s + (−1.51 − 0.873i)19-s + (−0.407 + 0.290i)20-s + (−0.109 − 0.0634i)22-s + (0.838 − 0.484i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.480751564\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.480751564\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.03 - 0.928i)T \) |
| 13 | \( 1 + (-1.10 + 3.43i)T \) |
good | 7 | \( 1 + (1.40 - 2.42i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.515 + 0.297i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.98 - 2.87i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.59 + 3.80i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.02 + 2.32i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.26 - 2.18i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.59iT - 31T^{2} \) |
| 37 | \( 1 + (-5.18 - 8.98i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.98 + 2.87i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.67 - 2.12i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 - 13.8iT - 53T^{2} \) |
| 59 | \( 1 + (8.40 + 4.85i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.41 - 5.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.93 - 6.80i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.11 - 0.642i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 - 4.19T + 83T^{2} \) |
| 89 | \( 1 + (-5.24 + 3.02i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.45 + 14.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.946564957902786757655972139481, −8.949789730414172753753437500936, −8.613446198040703643040934426162, −7.39581272895196647145854122551, −6.28995485349442912545222010313, −5.77994722341485865429051656080, −4.61177218232192961897195175304, −3.10471430992265145214096060231, −2.67359173218176866352013722905, −1.27090327490422101719201129700,
0.834356762703838226555041018965, 2.12498331431830334937418341172, 3.76567048411330762544412411307, 4.63177247703167859383488779207, 5.79893046263675921639487731515, 6.35061120782310967901823757889, 7.24640997539215001912882713119, 8.030137720495486505898494597421, 9.144702217368558178206582181952, 9.537355059008307237363778260695