| L(s) = 1 | + (0.0163 − 0.0104i)2-s + (0.749 − 5.21i)3-s + (−1.66 + 3.63i)4-s + (1.45 + 4.96i)5-s + (−0.0424 − 0.0929i)6-s + (−1.27 + 1.10i)7-s + (0.0220 + 0.153i)8-s + (−17.9 − 5.28i)9-s + (0.0758 + 0.0657i)10-s + (5.67 − 8.83i)11-s + (17.7 + 11.3i)12-s + (−8.57 + 9.90i)13-s + (−0.00924 + 0.0314i)14-s + (26.9 − 3.87i)15-s + (−10.4 − 12.0i)16-s + (14.9 − 6.83i)17-s + ⋯ |
| L(s) = 1 | + (0.00815 − 0.00524i)2-s + (0.249 − 1.73i)3-s + (−0.415 + 0.909i)4-s + (0.291 + 0.992i)5-s + (−0.00707 − 0.0154i)6-s + (−0.182 + 0.158i)7-s + (0.00275 + 0.0191i)8-s + (−1.99 − 0.586i)9-s + (0.00758 + 0.00657i)10-s + (0.516 − 0.803i)11-s + (1.47 + 0.949i)12-s + (−0.659 + 0.761i)13-s + (−0.000660 + 0.00224i)14-s + (1.79 − 0.258i)15-s + (−0.654 − 0.755i)16-s + (0.880 − 0.402i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.874920 - 0.240900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874920 - 0.240900i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-18.9 - 13.0i)T \) |
| good | 2 | \( 1 + (-0.0163 + 0.0104i)T + (1.66 - 3.63i)T^{2} \) |
| 3 | \( 1 + (-0.749 + 5.21i)T + (-8.63 - 2.53i)T^{2} \) |
| 5 | \( 1 + (-1.45 - 4.96i)T + (-21.0 + 13.5i)T^{2} \) |
| 7 | \( 1 + (1.27 - 1.10i)T + (6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (-5.67 + 8.83i)T + (-50.2 - 110. i)T^{2} \) |
| 13 | \( 1 + (8.57 - 9.90i)T + (-24.0 - 167. i)T^{2} \) |
| 17 | \( 1 + (-14.9 + 6.83i)T + (189. - 218. i)T^{2} \) |
| 19 | \( 1 + (20.8 + 9.52i)T + (236. + 272. i)T^{2} \) |
| 29 | \( 1 + (5.54 + 12.1i)T + (-550. + 635. i)T^{2} \) |
| 31 | \( 1 + (-1.70 - 11.8i)T + (-922. + 270. i)T^{2} \) |
| 37 | \( 1 + (5.02 - 17.1i)T + (-1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (-4.27 + 1.25i)T + (1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (-60.5 - 8.70i)T + (1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 + 29.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-19.2 + 16.6i)T + (399. - 2.78e3i)T^{2} \) |
| 59 | \( 1 + (11.7 - 13.5i)T + (-495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-7.59 + 1.09i)T + (3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (50.8 + 79.1i)T + (-1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (30.4 - 19.5i)T + (2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (3.42 - 7.49i)T + (-3.48e3 - 4.02e3i)T^{2} \) |
| 79 | \( 1 + (27.2 + 23.5i)T + (888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (-4.74 + 16.1i)T + (-5.79e3 - 3.72e3i)T^{2} \) |
| 89 | \( 1 + (13.2 + 1.89i)T + (7.60e3 + 2.23e3i)T^{2} \) |
| 97 | \( 1 + (-27.0 - 92.2i)T + (-7.91e3 + 5.08e3i)T^{2} \) |
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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
−17.74900038606629333561305888570, −16.84321031687120750754687799336, −14.47003952053709379779706109433, −13.67012417869756530302772911537, −12.54330962272813153797943476755, −11.46072484848221918425130853648, −8.980071726170932409987922937250, −7.53068713066876376379365586184, −6.50006285912538212167661749420, −2.84061606871502900895235607229,
4.34745341388436017684087423537, 5.46278428508676691994522962095, 8.785648738866592683133039795941, 9.750353856128983441979357013983, 10.56852341858789633626374300519, 12.73284759552765488305054364227, 14.52627467038480199246371766890, 15.08716797643187135319578908757, 16.49277147706101571690087029396, 17.33666950994179596562089763185
