| L(s) = 1 | + (0.540 + 0.841i)2-s + (−2.56 − 0.368i)3-s + (−0.415 + 0.909i)4-s + (1.88 − 1.19i)5-s + (−1.07 − 2.35i)6-s + (1.82 − 1.58i)7-s + (−0.989 + 0.142i)8-s + (3.56 + 1.04i)9-s + (2.02 + 0.941i)10-s + (1.25 + 0.808i)11-s + (1.40 − 2.17i)12-s + (3.06 + 2.65i)13-s + (2.31 + 0.680i)14-s + (−5.28 + 2.37i)15-s + (−0.654 − 0.755i)16-s + (5.62 − 2.56i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (−1.48 − 0.212i)3-s + (−0.207 + 0.454i)4-s + (0.844 − 0.535i)5-s + (−0.439 − 0.962i)6-s + (0.690 − 0.597i)7-s + (−0.349 + 0.0503i)8-s + (1.18 + 0.348i)9-s + (0.641 + 0.297i)10-s + (0.379 + 0.243i)11-s + (0.404 − 0.629i)12-s + (0.850 + 0.736i)13-s + (0.619 + 0.181i)14-s + (−1.36 + 0.612i)15-s + (−0.163 − 0.188i)16-s + (1.36 − 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15371 + 0.149931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15371 + 0.149931i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.88 + 1.19i)T \) |
| 23 | \( 1 + (3.56 - 3.21i)T \) |
| good | 3 | \( 1 + (2.56 + 0.368i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-1.82 + 1.58i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.25 - 0.808i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.06 - 2.65i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.62 + 2.56i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.291 + 0.638i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.53 + 5.54i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.649 + 4.51i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.60 - 5.45i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (4.89 - 1.43i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (4.99 + 0.718i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 4.52iT - 47T^{2} \) |
| 53 | \( 1 + (-4.01 + 3.48i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (9.64 - 11.1i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-2.04 - 14.2i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-1.77 - 2.75i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-9.51 + 6.11i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (3.05 + 1.39i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.54 + 1.77i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.823 + 2.80i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.515 - 3.58i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.34 - 11.3i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−11.96954424190066515556458440352, −11.70208495002036481976386366352, −10.41548128762205898922693016120, −9.429898190568182009367804084974, −8.022583932016483389133966159704, −6.88232567362407498072770937276, −5.94338123008011723855885421798, −5.22765886215756431836220042374, −4.16645990003599190522808013749, −1.34186112284228093948119279738,
1.54211775409847842789496895428, 3.47174861200468459928517325062, 5.18322678013473315778959875770, 5.69757796944959585858465391530, 6.56262435475224071987069469845, 8.341541416111939393112453914432, 9.732805277445511700093000918802, 10.67493436830766690042792057004, 11.03227341850019300686902513883, 12.13751254289935994082560132977
