| L(s) = 1 | + (0.918 + 1.78i)2-s + (−1.56 + 0.752i)3-s + (−1.17 + 1.64i)4-s + (2.94 − 2.80i)5-s + (−2.77 − 2.08i)6-s + (−1.97 − 1.75i)7-s + (−0.0375 − 0.00540i)8-s + (1.86 − 2.34i)9-s + (7.71 + 2.66i)10-s + (0.257 + 0.132i)11-s + (0.590 − 3.44i)12-s + (3.86 + 4.46i)13-s + (1.31 − 5.13i)14-s + (−2.48 + 6.59i)15-s + (1.29 + 3.74i)16-s + (2.07 − 0.198i)17-s + ⋯ |
| L(s) = 1 | + (0.649 + 1.25i)2-s + (−0.900 + 0.434i)3-s + (−0.585 + 0.822i)4-s + (1.31 − 1.25i)5-s + (−1.13 − 0.852i)6-s + (−0.747 − 0.664i)7-s + (−0.0132 − 0.00191i)8-s + (0.622 − 0.782i)9-s + (2.43 + 0.844i)10-s + (0.0775 + 0.0399i)11-s + (0.170 − 0.994i)12-s + (1.07 + 1.23i)13-s + (0.352 − 1.37i)14-s + (−0.641 + 1.70i)15-s + (0.323 + 0.935i)16-s + (0.504 − 0.0481i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58911 + 0.997175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58911 + 0.997175i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.56 - 0.752i)T \) |
| 7 | \( 1 + (1.97 + 1.75i)T \) |
| 23 | \( 1 + (3.40 - 3.38i)T \) |
| good | 2 | \( 1 + (-0.918 - 1.78i)T + (-1.16 + 1.62i)T^{2} \) |
| 5 | \( 1 + (-2.94 + 2.80i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (-0.257 - 0.132i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-3.86 - 4.46i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.07 + 0.198i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (-0.526 + 5.51i)T + (-18.6 - 3.59i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.284i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-6.65 + 2.66i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (3.31 + 0.804i)T + (32.8 + 16.9i)T^{2} \) |
| 41 | \( 1 + (0.0413 - 0.140i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (6.90 - 0.992i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (-1.41 - 0.816i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.27 + 1.40i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (-1.35 - 0.467i)T + (46.3 + 36.4i)T^{2} \) |
| 61 | \( 1 + (-3.70 + 4.71i)T + (-14.3 - 59.2i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 0.496i)T + (66.6 + 6.36i)T^{2} \) |
| 71 | \( 1 + (5.46 - 8.50i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.99 - 7.01i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-0.134 - 0.696i)T + (-73.3 + 29.3i)T^{2} \) |
| 83 | \( 1 + (-3.82 + 1.12i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (3.16 + 1.26i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (-1.17 + 3.98i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−11.15606196691106940676924939757, −9.978896090355555453433054429128, −9.442148697421813143689310255965, −8.430936385428972077606303749706, −6.89229464642087480680956217433, −6.35072922347960700314159825241, −5.60737771912202886852424231378, −4.76456191020517710509500305299, −3.97335885161218332328963607766, −1.31360619156842631053317560030,
1.53837227225630880634174072661, 2.67835652325747499999939324618, 3.54186582912399896658016832280, 5.33514769027391895495860685869, 5.99177918079623934869186767893, 6.64557871620849998962727481419, 8.064396093442193684979079781223, 9.778299160322025858897310736201, 10.35073967355464931723691379758, 10.68465568006870690113982372290
