| L(s) = 1 | + (−0.136 − 0.141i)2-s + (1.95 + 2.58i)3-s + (0.0603 − 1.96i)4-s + (−0.282 − 0.802i)5-s + (0.0977 − 0.629i)6-s + (0.255 − 1.16i)7-s + (−0.575 + 0.524i)8-s + (−2.04 + 7.20i)9-s + (−0.0745 + 0.149i)10-s + (−4.29 + 1.08i)11-s + (5.18 − 3.66i)12-s + (1.87 + 1.71i)13-s + (−0.200 + 0.123i)14-s + (1.52 − 2.29i)15-s + (−3.76 − 0.232i)16-s + (−1.00 − 6.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.0968 − 0.0998i)2-s + (1.12 + 1.49i)3-s + (0.0301 − 0.980i)4-s + (−0.126 − 0.358i)5-s + (0.0398 − 0.256i)6-s + (0.0967 − 0.441i)7-s + (−0.203 + 0.185i)8-s + (−0.682 + 2.40i)9-s + (−0.0235 + 0.0473i)10-s + (−1.29 + 0.326i)11-s + (1.49 − 1.05i)12-s + (0.520 + 0.474i)13-s + (−0.0534 + 0.0331i)14-s + (0.392 − 0.592i)15-s + (−0.940 − 0.0580i)16-s + (−0.243 − 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21193 + 0.247966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21193 + 0.247966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (-8.14 - 6.06i)T \) |
| good | 2 | \( 1 + (0.136 + 0.141i)T + (-0.0615 + 1.99i)T^{2} \) |
| 3 | \( 1 + (-1.95 - 2.58i)T + (-0.820 + 2.88i)T^{2} \) |
| 5 | \( 1 + (0.282 + 0.802i)T + (-3.89 + 3.13i)T^{2} \) |
| 7 | \( 1 + (-0.255 + 1.16i)T + (-6.35 - 2.92i)T^{2} \) |
| 11 | \( 1 + (4.29 - 1.08i)T + (9.69 - 5.20i)T^{2} \) |
| 13 | \( 1 + (-1.87 - 1.71i)T + (1.19 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.00 + 6.46i)T + (-16.2 + 5.15i)T^{2} \) |
| 19 | \( 1 + (1.34 + 3.17i)T + (-13.2 + 13.6i)T^{2} \) |
| 23 | \( 1 + (-1.79 - 6.31i)T + (-19.5 + 12.1i)T^{2} \) |
| 29 | \( 1 + (0.764 - 0.891i)T + (-4.44 - 28.6i)T^{2} \) |
| 31 | \( 1 + (0.0100 + 0.0201i)T + (-18.6 + 24.7i)T^{2} \) |
| 37 | \( 1 + (0.537 + 5.79i)T + (-36.3 + 6.79i)T^{2} \) |
| 41 | \( 1 + (0.750 - 2.12i)T + (-31.9 - 25.7i)T^{2} \) |
| 43 | \( 1 + (-2.58 - 1.83i)T + (14.2 + 40.5i)T^{2} \) |
| 47 | \( 1 + (-3.98 - 6.90i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.00 + 0.620i)T + (51.3 + 12.9i)T^{2} \) |
| 59 | \( 1 + (-0.0489 - 0.223i)T + (-53.5 + 24.6i)T^{2} \) |
| 61 | \( 1 + (-11.5 - 4.49i)T + (45.0 + 41.0i)T^{2} \) |
| 67 | \( 1 + (14.9 + 4.74i)T + (54.6 + 38.7i)T^{2} \) |
| 71 | \( 1 + (-1.37 - 1.60i)T + (-10.8 + 70.1i)T^{2} \) |
| 73 | \( 1 + (7.22 - 1.35i)T + (68.0 - 26.3i)T^{2} \) |
| 79 | \( 1 + (-5.64 - 1.05i)T + (73.6 + 28.5i)T^{2} \) |
| 83 | \( 1 + (-10.5 + 3.35i)T + (67.7 - 47.9i)T^{2} \) |
| 89 | \( 1 + (3.33 - 2.06i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (1.28 - 8.28i)T + (-92.4 - 29.4i)T^{2} \) |
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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
−14.00235840938245387392426534294, −13.36784290974146554010743320108, −11.20691431744621362868044967283, −10.53744565287141432708329162019, −9.505864971516414386135753377935, −8.903149052821192751031185162357, −7.47144884375435813032766379505, −5.24747317076223185636877154884, −4.45581610088547585488028977557, −2.67359576158406938862025159567,
2.37494697929132283949202640333, 3.47008509792866343096411994196, 6.18721010564161657630614750791, 7.32492084604079708751119528603, 8.357812591472912053697032653665, 8.575926917063703499003186637495, 10.66217771241998024763594950130, 12.15075502996959716628148393403, 12.88820104653025903409527465013, 13.38987281249822862938654775984
