| L(s) = 1 | + (1.49 − 0.375i)2-s + (0.518 + 0.686i)3-s + (0.322 − 0.173i)4-s + (1.05 − 1.22i)5-s + (1.03 + 0.829i)6-s + (−3.19 + 1.01i)7-s + (−1.85 + 1.69i)8-s + (0.618 − 2.17i)9-s + (1.10 − 2.22i)10-s + (−0.667 − 0.688i)11-s + (0.286 + 0.131i)12-s + (1.51 + 1.38i)13-s + (−4.38 + 2.71i)14-s + (1.38 + 0.0855i)15-s + (−2.54 + 3.83i)16-s + (2.68 − 2.16i)17-s + ⋯ |
| L(s) = 1 | + (1.05 − 0.265i)2-s + (0.299 + 0.396i)3-s + (0.161 − 0.0866i)4-s + (0.470 − 0.548i)5-s + (0.420 + 0.338i)6-s + (−1.20 + 0.384i)7-s + (−0.656 + 0.598i)8-s + (0.206 − 0.724i)9-s + (0.350 − 0.703i)10-s + (−0.201 − 0.207i)11-s + (0.0826 + 0.0380i)12-s + (0.420 + 0.383i)13-s + (−1.17 + 0.726i)14-s + (0.358 + 0.0220i)15-s + (−0.635 + 0.958i)16-s + (0.651 − 0.524i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58983 - 0.0559465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58983 - 0.0559465i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (8.58 - 5.41i)T \) |
| good | 2 | \( 1 + (-1.49 + 0.375i)T + (1.76 - 0.946i)T^{2} \) |
| 3 | \( 1 + (-0.518 - 0.686i)T + (-0.820 + 2.88i)T^{2} \) |
| 5 | \( 1 + (-1.05 + 1.22i)T + (-0.766 - 4.94i)T^{2} \) |
| 7 | \( 1 + (3.19 - 1.01i)T + (5.71 - 4.04i)T^{2} \) |
| 11 | \( 1 + (0.667 + 0.688i)T + (-0.338 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.51 - 1.38i)T + (1.19 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.68 + 2.16i)T + (3.63 - 16.6i)T^{2} \) |
| 19 | \( 1 + (5.75 + 0.712i)T + (18.4 + 4.63i)T^{2} \) |
| 23 | \( 1 + (0.591 + 2.07i)T + (-19.5 + 12.1i)T^{2} \) |
| 29 | \( 1 + (-2.66 - 7.55i)T + (-22.5 + 18.1i)T^{2} \) |
| 31 | \( 1 + (-2.11 - 4.24i)T + (-18.6 + 24.7i)T^{2} \) |
| 37 | \( 1 + (0.993 + 10.7i)T + (-36.3 + 6.79i)T^{2} \) |
| 41 | \( 1 + (-0.589 - 0.688i)T + (-6.28 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-8.41 + 3.87i)T + (27.9 - 32.6i)T^{2} \) |
| 47 | \( 1 + (-0.555 + 0.962i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.69 - 6.37i)T + (-36.8 + 38.0i)T^{2} \) |
| 59 | \( 1 + (5.71 + 1.81i)T + (48.1 + 34.0i)T^{2} \) |
| 61 | \( 1 + (-1.41 - 0.548i)T + (45.0 + 41.0i)T^{2} \) |
| 67 | \( 1 + (2.08 + 9.53i)T + (-60.8 + 28.0i)T^{2} \) |
| 71 | \( 1 + (2.23 - 6.34i)T + (-55.3 - 44.5i)T^{2} \) |
| 73 | \( 1 + (8.02 - 1.50i)T + (68.0 - 26.3i)T^{2} \) |
| 79 | \( 1 + (-10.8 - 2.03i)T + (73.6 + 28.5i)T^{2} \) |
| 83 | \( 1 + (1.21 - 5.54i)T + (-75.4 - 34.6i)T^{2} \) |
| 89 | \( 1 + (-1.66 + 1.03i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (14.7 + 11.8i)T + (20.7 + 94.7i)T^{2} \) |
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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
−13.72483571000074225092479298251, −12.57639607540451203889316575630, −12.41398469091323329718070901928, −10.68483699274936593700874256314, −9.275404714098502391964259008140, −8.823506622148983165950812539317, −6.57556635924921396171547382410, −5.50217843397004644715514491730, −4.09127233768869882212386886491, −2.93772201247181125810244732980,
2.80142683740518611494728842970, 4.23632874808368974179850126451, 5.93601748567721879161941542012, 6.64969366316358489159362533138, 8.082231657190478599665108504328, 9.773842651279689968571480228082, 10.48408770368436112456340484866, 12.27148436603038875583539282432, 13.27384686033490211419952303974, 13.51007336792732986762608235696
