| L(s) = 1 | + (−0.0125 − 0.0872i)2-s + (−0.415 + 0.909i)3-s + (1.91 − 0.561i)4-s + (0.654 − 0.755i)5-s + (0.0845 + 0.0248i)6-s + (−1.30 + 0.835i)7-s + (−0.146 − 0.320i)8-s + (1.30 + 1.51i)9-s + (−0.0741 − 0.0476i)10-s + (0.289 − 2.01i)11-s + (−0.283 + 1.97i)12-s + (1.80 + 1.15i)13-s + (0.0892 + 0.103i)14-s + (0.415 + 0.909i)15-s + (3.32 − 2.13i)16-s + (−4.18 − 1.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.00887 − 0.0617i)2-s + (−0.239 + 0.525i)3-s + (0.955 − 0.280i)4-s + (0.292 − 0.337i)5-s + (0.0345 + 0.0101i)6-s + (−0.491 + 0.315i)7-s + (−0.0516 − 0.113i)8-s + (0.436 + 0.503i)9-s + (−0.0234 − 0.0150i)10-s + (0.0872 − 0.607i)11-s + (−0.0818 + 0.569i)12-s + (0.499 + 0.320i)13-s + (0.0238 + 0.0275i)14-s + (0.107 + 0.234i)15-s + (0.831 − 0.534i)16-s + (−1.01 − 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15923 + 0.0908989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15923 + 0.0908989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (2.79 + 3.89i)T \) |
| good | 2 | \( 1 + (0.0125 + 0.0872i)T + (-1.91 + 0.563i)T^{2} \) |
| 3 | \( 1 + (0.415 - 0.909i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (1.30 - 0.835i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.289 + 2.01i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.80 - 1.15i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (4.18 + 1.22i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (7.12 - 2.09i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (4.30 + 1.26i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.376 + 0.824i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-7.26 - 8.38i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.95 + 4.56i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.24 + 4.90i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 2.72T + 47T^{2} \) |
| 53 | \( 1 + (0.230 - 0.148i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.74 - 4.33i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (1.33 + 2.92i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.279 - 1.94i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.58 - 11.0i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-9.95 + 2.92i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.70 - 1.74i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.85 - 3.29i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.266 + 0.584i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.62 + 4.18i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−13.48653549201664269877939336081, −12.54124064681176938963109322654, −11.27211327147402442998720557591, −10.60098062629977620297354407520, −9.548859945299467283738260002782, −8.284187376684876144218778520808, −6.65323393467353746571251657089, −5.80596101824442946334817394637, −4.20981357212134820057600913110, −2.22148551717071246727612662844,
2.08479681043746701826867835677, 3.88601739345142323769478183359, 6.16193615712744591226598464525, 6.72846889982137943700087560372, 7.77624647921809038641159152382, 9.375976034954742091532088734218, 10.62315223331054397828603004175, 11.42614149265873479217641666473, 12.75956822998713490096773595111, 13.08763686385752919128554360715
