| L(s) = 1 | + (−0.540 + 0.841i)2-s + (−2.80 + 0.402i)3-s + (−0.415 − 0.909i)4-s + (1.39 − 1.74i)5-s + (1.17 − 2.57i)6-s + (2.07 + 1.79i)7-s + (0.989 + 0.142i)8-s + (4.81 − 1.41i)9-s + (0.719 + 2.11i)10-s + (−2.81 + 1.80i)11-s + (1.53 + 2.38i)12-s + (−2.48 + 2.15i)13-s + (−2.63 + 0.774i)14-s + (−3.19 + 5.46i)15-s + (−0.654 + 0.755i)16-s + (3.09 + 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.594i)2-s + (−1.61 + 0.232i)3-s + (−0.207 − 0.454i)4-s + (0.622 − 0.782i)5-s + (0.480 − 1.05i)6-s + (0.785 + 0.680i)7-s + (0.349 + 0.0503i)8-s + (1.60 − 0.471i)9-s + (0.227 + 0.669i)10-s + (−0.848 + 0.545i)11-s + (0.441 + 0.687i)12-s + (−0.689 + 0.597i)13-s + (−0.704 + 0.206i)14-s + (−0.825 + 1.41i)15-s + (−0.163 + 0.188i)16-s + (0.751 + 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0150 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0150 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.449656 + 0.456489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.449656 + 0.456489i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (-1.39 + 1.74i)T \) |
| 23 | \( 1 + (-4.50 - 1.65i)T \) |
| good | 3 | \( 1 + (2.80 - 0.402i)T + (2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-2.07 - 1.79i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (2.81 - 1.80i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.48 - 2.15i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.09 - 1.41i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.40 - 7.46i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.0669 - 0.146i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.0425 - 0.296i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.77 - 9.45i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.66 - 1.07i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (5.68 - 0.816i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 11.2iT - 47T^{2} \) |
| 53 | \( 1 + (-3.34 - 2.89i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.88 - 3.32i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.444 + 3.09i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.45 - 5.37i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (7.55 + 4.85i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (3.43 - 1.56i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.12 - 4.75i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (4.60 + 15.6i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.04 + 7.27i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (0.893 - 3.04i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−12.15429304825155754354598264977, −11.66588794444850375403932222396, −10.24537392217203308207154282525, −9.834561498440126289572692278243, −8.479685412336410768821798258892, −7.37171167013124427595697102789, −5.99861597711223471676400789507, −5.31707175306057196224594883327, −4.73474118142598858576714453255, −1.52903905307078829828229951206,
0.807362032621069725796264344134, 2.78876777707130260388702486549, 4.84603556975193229583029995900, 5.60227087548164456436724125676, 7.02303295857767062393765213851, 7.66256886707294190829899101793, 9.460739659539010183590845524971, 10.53473920796790201520884671591, 10.93241425965036957429948629300, 11.56948216572435912355221705358
