| L(s) = 1 | + (−0.593 + 0.804i)2-s + (2.32 − 0.439i)3-s + (−0.294 − 0.955i)4-s + (1.84 − 1.26i)5-s + (−1.02 + 2.13i)6-s + (2.34 + 1.22i)7-s + (0.943 + 0.330i)8-s + (2.41 − 0.947i)9-s + (−0.0793 + 2.23i)10-s + (2.25 − 5.74i)11-s + (−1.10 − 2.09i)12-s + (−3.25 − 0.366i)13-s + (−2.37 + 1.15i)14-s + (3.73 − 3.74i)15-s + (−0.826 + 0.563i)16-s + (−5.69 − 0.213i)17-s + ⋯ |
| L(s) = 1 | + (−0.419 + 0.568i)2-s + (1.34 − 0.253i)3-s + (−0.147 − 0.477i)4-s + (0.825 − 0.564i)5-s + (−0.418 + 0.869i)6-s + (0.885 + 0.464i)7-s + (0.333 + 0.116i)8-s + (0.804 − 0.315i)9-s + (−0.0251 + 0.706i)10-s + (0.679 − 1.73i)11-s + (−0.319 − 0.603i)12-s + (−0.902 − 0.101i)13-s + (−0.636 + 0.309i)14-s + (0.963 − 0.967i)15-s + (−0.206 + 0.140i)16-s + (−1.38 − 0.0517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04914 - 0.00524048i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04914 - 0.00524048i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.593 - 0.804i)T \) |
| 5 | \( 1 + (-1.84 + 1.26i)T \) |
| 7 | \( 1 + (-2.34 - 1.22i)T \) |
| good | 3 | \( 1 + (-2.32 + 0.439i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-2.25 + 5.74i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (3.25 + 0.366i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (5.69 + 0.213i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (3.42 - 5.93i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.103 - 2.77i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (0.586 - 0.133i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.39 + 1.96i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.96 - 3.15i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-3.24 - 6.73i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.524 + 1.50i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-8.61 - 6.35i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-6.94 - 3.67i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.0626 - 0.835i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.906 + 2.93i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.0686 - 0.0184i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.988 + 4.33i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (4.91 - 3.62i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (6.24 + 3.60i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.73 - 15.4i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (5.99 + 15.2i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-8.90 - 8.90i)T + 97iT^{2} \) |
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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
−10.76660481953204525935854878183, −9.630731247391976618946178275279, −8.760865629029796955510301077674, −8.559342715209043914694749592169, −7.68905071502228781841368011149, −6.34233548203115327939040112784, −5.52327857111674461376561922653, −4.21896373392312831020593046487, −2.60670465508352922893372181959, −1.53504419894739130205258133968,
2.11038406346898510895688996549, 2.36115339978367073166401232301, 4.06419396316732995415327717128, 4.78695481574318065890757159093, 6.91649835169682203949597480615, 7.28648040637343542722094849397, 8.716430792271753399752268584401, 9.098757986170683901982922214533, 10.04796520807092488961305819894, 10.62422799071399449097050024078
