| L(s) = 1 | + (−0.226 + 0.0666i)2-s + (−0.313 − 0.361i)3-s + (−1.63 + 1.05i)4-s + (−0.215 − 1.49i)5-s + (0.0952 + 0.0612i)6-s + (−1.05 + 2.31i)7-s + (0.610 − 0.704i)8-s + (0.394 − 2.74i)9-s + (0.148 + 0.325i)10-s + (3.23 + 0.950i)11-s + (0.893 + 0.262i)12-s + (1.36 + 2.99i)13-s + (0.0856 − 0.595i)14-s + (−0.474 + 0.547i)15-s + (1.52 − 3.33i)16-s + (−5.28 − 3.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.160 + 0.0471i)2-s + (−0.181 − 0.208i)3-s + (−0.817 + 0.525i)4-s + (−0.0963 − 0.669i)5-s + (0.0388 + 0.0249i)6-s + (−0.399 + 0.875i)7-s + (0.215 − 0.249i)8-s + (0.131 − 0.914i)9-s + (0.0470 + 0.102i)10-s + (0.975 + 0.286i)11-s + (0.257 + 0.0757i)12-s + (0.379 + 0.830i)13-s + (0.0229 − 0.159i)14-s + (−0.122 + 0.141i)15-s + (0.380 − 0.834i)16-s + (−1.28 − 0.823i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.514473 + 0.00301407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.514473 + 0.00301407i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (4.72 - 0.847i)T \) |
| good | 2 | \( 1 + (0.226 - 0.0666i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (0.313 + 0.361i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.215 + 1.49i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.05 - 2.31i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.23 - 0.950i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 2.99i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.28 + 3.39i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.55 - 2.28i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.28 - 3.39i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.10 + 3.58i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.00540 - 0.0375i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.462 + 3.21i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.844 + 0.974i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 + (2.79 - 6.11i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (4.32 + 9.47i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.35 - 2.72i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-6.54 + 1.92i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (3.92 - 1.15i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.66 + 2.35i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.997 - 2.18i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.420 - 2.92i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (8.09 + 9.34i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.51 - 10.5i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−17.94951643430532934827866971651, −16.90247465795691121675820141621, −15.67479718618704487897711942866, −14.04141664047054737774515654298, −12.60358819790171675590892569951, −11.91810803168186069586767540613, −9.375542029214594767324407564319, −8.712922685644967616747399953752, −6.51776563858356326443205144895, −4.27602166279250101438312391251,
4.28468710727745230810281531166, 6.43377255067923003768276281752, 8.448140238748239014236300165545, 10.21495512329415600966028170603, 10.91068771284362341055243992990, 13.16736683908858216552996279976, 14.08028925293663045890794494363, 15.41461603707356350799396286112, 16.93899123217625348440746094667, 17.91269484645337483769851496060
