| L(s) = 1 | + (−0.691 − 2.06i)2-s + (−1.31 + 0.740i)3-s + (−2.18 + 1.65i)4-s + (−2.30 − 3.47i)5-s + (2.43 + 2.20i)6-s + (0.363 + 0.802i)7-s + (1.33 + 0.915i)8-s + (−0.373 + 0.612i)9-s + (−5.57 + 7.15i)10-s + (−1.49 + 0.0955i)11-s + (1.66 − 3.79i)12-s + (0.264 − 0.306i)13-s + (1.40 − 1.30i)14-s + (5.60 + 2.87i)15-s + (−0.538 + 1.88i)16-s + (−0.645 − 0.285i)17-s + ⋯ |
| L(s) = 1 | + (−0.488 − 1.46i)2-s + (−0.760 + 0.427i)3-s + (−1.09 + 0.825i)4-s + (−1.02 − 1.55i)5-s + (0.995 + 0.901i)6-s + (0.137 + 0.303i)7-s + (0.471 + 0.323i)8-s + (−0.124 + 0.204i)9-s + (−1.76 + 2.26i)10-s + (−0.449 + 0.0288i)11-s + (0.479 − 1.09i)12-s + (0.0734 − 0.0848i)13-s + (0.375 − 0.349i)14-s + (1.44 + 0.741i)15-s + (−0.134 + 0.471i)16-s + (−0.156 − 0.0691i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.309345 - 0.107433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.309345 - 0.107433i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (31.3 + 1.54i)T \) |
| good | 2 | \( 1 + (0.691 + 2.06i)T + (-1.59 + 1.20i)T^{2} \) |
| 3 | \( 1 + (1.31 - 0.740i)T + (1.56 - 2.56i)T^{2} \) |
| 5 | \( 1 + (2.30 + 3.47i)T + (-1.94 + 4.60i)T^{2} \) |
| 7 | \( 1 + (-0.363 - 0.802i)T + (-4.61 + 5.26i)T^{2} \) |
| 11 | \( 1 + (1.49 - 0.0955i)T + (10.9 - 1.40i)T^{2} \) |
| 13 | \( 1 + (-0.264 + 0.306i)T + (-1.86 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.645 + 0.285i)T + (11.4 + 12.5i)T^{2} \) |
| 19 | \( 1 + (4.63 - 3.54i)T + (4.98 - 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.765 + 7.22i)T + (-22.4 - 4.82i)T^{2} \) |
| 29 | \( 1 + (-5.18 - 7.49i)T + (-10.2 + 27.1i)T^{2} \) |
| 31 | \( 1 + (-0.728 + 1.08i)T + (-11.7 - 28.7i)T^{2} \) |
| 37 | \( 1 + (3.37 + 0.838i)T + (32.7 + 17.2i)T^{2} \) |
| 41 | \( 1 + (-3.64 + 0.327i)T + (40.3 - 7.30i)T^{2} \) |
| 43 | \( 1 + (1.36 - 1.87i)T + (-13.1 - 40.9i)T^{2} \) |
| 47 | \( 1 + (1.87 - 6.26i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (6.30 + 5.78i)T + (4.57 + 52.8i)T^{2} \) |
| 59 | \( 1 + (4.44 - 0.341i)T + (58.3 - 9.02i)T^{2} \) |
| 61 | \( 1 + (-5.53 + 4.69i)T + (9.90 - 60.1i)T^{2} \) |
| 67 | \( 1 + (-6.71 - 5.77i)T + (10.0 + 66.2i)T^{2} \) |
| 71 | \( 1 + (2.96 + 2.42i)T + (14.2 + 69.5i)T^{2} \) |
| 73 | \( 1 + (-11.9 + 8.57i)T + (23.6 - 69.0i)T^{2} \) |
| 79 | \( 1 + (-1.80 - 0.267i)T + (75.6 + 22.9i)T^{2} \) |
| 83 | \( 1 + (-7.05 - 2.18i)T + (68.4 + 46.9i)T^{2} \) |
| 89 | \( 1 + (-5.07 + 3.33i)T + (35.1 - 81.7i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 0.549i)T + (96.4 + 9.91i)T^{2} \) |
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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
−10.21243439394444103516314287340, −9.117993724205675387320028564254, −8.470419688011341552784498077830, −8.017556802896626080106496395695, −6.31698594016167386509842707542, −5.03143604598121129286798653134, −4.59260134910802528144326673001, −3.59930554210483276465016927618, −2.17828800367097125162399963971, −0.78281440496656498650795152698,
0.29857205933894798669662779313, 2.75968719114508741545075882472, 4.01130057424110293484044110861, 5.25833213631130041257551478330, 6.39316455728316930290834761493, 6.60746124857414749496124874762, 7.47380123732908890413759642490, 7.949830046040144890896267772060, 8.950243830475223812995666097117, 10.09738163616239197594864765005
