| L(s) = 1 | + (0.198 − 0.435i)2-s + (−2.11 − 0.620i)3-s + (1.15 + 1.33i)4-s + (−2.18 + 1.40i)5-s + (−0.691 + 0.797i)6-s + (0.483 − 3.36i)7-s + (1.73 − 0.508i)8-s + (1.56 + 1.00i)9-s + (0.176 + 1.22i)10-s + (0.0950 + 0.208i)11-s + (−1.62 − 3.54i)12-s + (0.435 + 3.02i)13-s + (−1.36 − 0.879i)14-s + (5.48 − 1.61i)15-s + (−0.380 + 2.64i)16-s + (1.26 − 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.140 − 0.308i)2-s + (−1.22 − 0.358i)3-s + (0.579 + 0.669i)4-s + (−0.976 + 0.627i)5-s + (−0.282 + 0.325i)6-s + (0.182 − 1.27i)7-s + (0.612 − 0.179i)8-s + (0.520 + 0.334i)9-s + (0.0559 + 0.388i)10-s + (0.0286 + 0.0627i)11-s + (−0.467 − 1.02i)12-s + (0.120 + 0.839i)13-s + (−0.365 − 0.235i)14-s + (1.41 − 0.415i)15-s + (−0.0952 + 0.662i)16-s + (0.306 − 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.522645 - 0.0741564i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.522645 - 0.0741564i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (4.62 - 1.25i)T \) |
| good | 2 | \( 1 + (-0.198 + 0.435i)T + (-1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (2.11 + 0.620i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (2.18 - 1.40i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.483 + 3.36i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.0950 - 0.208i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.435 - 3.02i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 1.45i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.26 + 1.46i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.23 + 4.89i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (1.44 - 0.424i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (5.67 + 3.64i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-6.78 + 4.36i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (2.55 + 0.749i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 + (1.22 - 8.49i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.00878 + 0.0611i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.0426 + 0.0125i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-5.15 + 11.2i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.46 - 7.58i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.437 - 0.505i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.70 - 11.8i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (0.303 + 0.194i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-15.4 - 4.54i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (0.335 - 0.215i)T + (40.2 - 88.2i)T^{2} \) |
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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
−17.67124850837853196870804823785, −16.77549438280340543921186921433, −15.80233741763860238681677209725, −13.91452657831320887142581353274, −12.27355814871927646413260670117, −11.43344959459257525705111511349, −10.66371080744931333057356507937, −7.62411840852442737605105286610, −6.71610085194056859926149333211, −4.02744855059760795335469833124,
5.02391178006753994249697831566, 6.09176650110013088563725642668, 8.245259285711677999907873202935, 10.37030895406267864126384531529, 11.60364304372605875531843957745, 12.37142508458322437423924466078, 14.76213691508780106542578001162, 15.82622556652173074562931536094, 16.34346305871690337678584639273, 17.86050679187521477445319340182
