| L(s) = 1 | + (2.73 − 0.732i)2-s + (−0.984 + 0.568i)3-s + (6.92 − 4i)4-s + (−27.3 − 7.32i)5-s + (−2.27 + 2.27i)6-s + (−37.5 − 65.0i)7-s + (15.9 − 16i)8-s + (−39.8 + 69.0i)9-s − 80.0·10-s − 183. i·11-s + (−4.54 + 7.87i)12-s + (82.1 + 22.0i)13-s + (−150. − 150. i)14-s + (31.0 − 8.32i)15-s + (31.9 − 55.4i)16-s + (7.71 + 28.7i)17-s + ⋯ |
| L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.109 + 0.0631i)3-s + (0.433 − 0.250i)4-s + (−1.09 − 0.293i)5-s + (−0.0631 + 0.0631i)6-s + (−0.766 − 1.32i)7-s + (0.249 − 0.250i)8-s + (−0.492 + 0.852i)9-s − 0.800·10-s − 1.51i·11-s + (−0.0315 + 0.0547i)12-s + (0.486 + 0.130i)13-s + (−0.766 − 0.766i)14-s + (0.138 − 0.0370i)15-s + (0.124 − 0.216i)16-s + (0.0266 + 0.0995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.439239 - 1.10506i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.439239 - 1.10506i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.73 + 0.732i)T \) |
| 37 | \( 1 + (873. + 1.05e3i)T \) |
| good | 3 | \( 1 + (0.984 - 0.568i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (27.3 + 7.32i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (37.5 + 65.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + 183. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-82.1 - 22.0i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-7.71 - 28.7i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (221. + 59.4i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (7.30 - 7.30i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-251. - 251. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-581. - 581. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-2.22e3 + 1.28e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-294. + 294. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.45e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.45e3 - 2.51e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.05e3 + 3.94e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (747. - 2.78e3i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-3.53e3 + 2.04e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (1.76e3 + 3.05e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + 8.73e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.11e4 + 2.97e3i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-3.79e3 + 6.56e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.15e4 - 3.10e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (3.84e3 - 3.84e3i)T - 8.85e7iT^{2} \) |
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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
−13.56898494516853936305667776119, −12.39025261019183262616758725157, −11.08511876645776278834425010306, −10.62287163267782287437162441938, −8.626512215366496672992447206046, −7.44221393283807310954067654810, −6.01965897342251806833856439674, −4.34187835174673071023888348739, −3.31296104568936054824715635069, −0.49017294748975554704439941923,
2.76248117554221318085664799887, 4.16110701679334389779004224388, 5.87375472324971797440805670774, 6.93629946165950325373487703646, 8.363680164124332061274471667075, 9.697317948152098691380647028511, 11.42263335555118114879415207167, 12.19268686633172221820792867296, 12.81272765716547345417647173996, 14.55307362148976337103474847676
