| L(s) = 1 | + (1.80 + 0.336i)2-s + (0.0591 − 0.638i)3-s + (1.26 + 0.489i)4-s + (−0.551 + 0.730i)5-s + (0.321 − 1.13i)6-s + (−0.608 + 0.376i)7-s + (−1.00 − 0.620i)8-s + (2.54 + 0.475i)9-s + (−1.23 + 1.12i)10-s + (−4.30 − 0.804i)11-s + (0.387 − 0.778i)12-s + (−0.590 + 0.365i)13-s + (−1.22 + 0.473i)14-s + (0.433 + 0.395i)15-s + (−3.60 − 3.28i)16-s + (0.124 + 0.437i)17-s + ⋯ |
| L(s) = 1 | + (1.27 + 0.238i)2-s + (0.0341 − 0.368i)3-s + (0.632 + 0.244i)4-s + (−0.246 + 0.326i)5-s + (0.131 − 0.461i)6-s + (−0.230 + 0.142i)7-s + (−0.354 − 0.219i)8-s + (0.848 + 0.158i)9-s + (−0.391 + 0.357i)10-s + (−1.29 − 0.242i)11-s + (0.111 − 0.224i)12-s + (−0.163 + 0.101i)13-s + (−0.326 + 0.126i)14-s + (0.111 + 0.102i)15-s + (−0.900 − 0.820i)16-s + (0.0301 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64702 + 0.0690598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64702 + 0.0690598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (9.77 - 2.71i)T \) |
| good | 2 | \( 1 + (-1.80 - 0.336i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (-0.0591 + 0.638i)T + (-2.94 - 0.551i)T^{2} \) |
| 5 | \( 1 + (0.551 - 0.730i)T + (-1.36 - 4.80i)T^{2} \) |
| 7 | \( 1 + (0.608 - 0.376i)T + (3.12 - 6.26i)T^{2} \) |
| 11 | \( 1 + (4.30 + 0.804i)T + (10.2 + 3.97i)T^{2} \) |
| 13 | \( 1 + (0.590 - 0.365i)T + (5.79 - 11.6i)T^{2} \) |
| 17 | \( 1 + (-0.124 - 0.437i)T + (-14.4 + 8.94i)T^{2} \) |
| 19 | \( 1 + (-0.187 - 2.01i)T + (-18.6 + 3.49i)T^{2} \) |
| 23 | \( 1 + (-1.85 + 0.346i)T + (21.4 - 8.30i)T^{2} \) |
| 29 | \( 1 + (-3.61 + 4.78i)T + (-7.93 - 27.8i)T^{2} \) |
| 31 | \( 1 + (-2.66 - 2.42i)T + (2.86 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-4.15 + 8.33i)T + (-22.2 - 29.5i)T^{2} \) |
| 41 | \( 1 + (-1.58 - 2.09i)T + (-11.2 + 39.4i)T^{2} \) |
| 43 | \( 1 + (-0.342 - 0.688i)T + (-25.9 + 34.3i)T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 + (0.0608 + 0.656i)T + (-52.0 + 9.73i)T^{2} \) |
| 59 | \( 1 + (-0.294 - 0.182i)T + (26.2 + 52.8i)T^{2} \) |
| 61 | \( 1 + (-1.70 - 6.00i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (5.01 + 3.10i)T + (29.8 + 59.9i)T^{2} \) |
| 71 | \( 1 + (7.34 + 9.72i)T + (-19.4 + 68.2i)T^{2} \) |
| 73 | \( 1 + (-9.14 - 12.1i)T + (-19.9 + 70.2i)T^{2} \) |
| 79 | \( 1 + (-6.35 + 8.41i)T + (-21.6 - 75.9i)T^{2} \) |
| 83 | \( 1 + (7.57 - 4.69i)T + (36.9 - 74.2i)T^{2} \) |
| 89 | \( 1 + (14.7 - 5.71i)T + (65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (-3.64 + 12.8i)T + (-82.4 - 51.0i)T^{2} \) |
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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
−13.64705214833770274117433899640, −12.96549457329537874728372979501, −12.22331779852484402454867878939, −10.88482417232809433398504509736, −9.648251179485930088145472762126, −7.927212742017750289390364787697, −6.85446580255965978435669082824, −5.63493847037574777230314605857, −4.37977662356969390930238433924, −2.86573010422304388525115718459,
2.91629091361381689111435770311, 4.38681721790014190505723335766, 5.16538336492268333977315948350, 6.76940522889664844190496423828, 8.268772138292976701550285036487, 9.725929641281215873955675816891, 10.79156661750322629132153163045, 12.08663695548575988357115107319, 12.89344038247570000839198483374, 13.52408291936496651719300535324
