| L(s) = 1 | + (0.974 − 0.222i)2-s + (0.875 + 0.483i)3-s + (0.900 − 0.433i)4-s + (−1.31 − 2.37i)5-s + (0.960 + 0.276i)6-s + (−2.30 − 2.88i)7-s + (0.781 − 0.623i)8-s + (0.532 + 0.846i)9-s + (−1.80 − 2.02i)10-s + (−1.04 + 0.364i)11-s + (0.998 + 0.0560i)12-s + (0.295 − 2.62i)13-s + (−2.88 − 2.30i)14-s − 2.71i·15-s + (0.623 − 0.781i)16-s + (−1.72 + 1.22i)17-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.157i)2-s + (0.505 + 0.279i)3-s + (0.450 − 0.216i)4-s + (−0.586 − 1.06i)5-s + (0.392 + 0.113i)6-s + (−0.869 − 1.09i)7-s + (0.276 − 0.220i)8-s + (0.177 + 0.282i)9-s + (−0.571 − 0.639i)10-s + (−0.314 + 0.110i)11-s + (0.288 + 0.0161i)12-s + (0.0819 − 0.727i)13-s + (−0.771 − 0.615i)14-s − 0.700i·15-s + (0.155 − 0.195i)16-s + (−0.417 + 0.296i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28981 - 1.51222i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28981 - 1.51222i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.974 + 0.222i)T \) |
| 3 | \( 1 + (-0.875 - 0.483i)T \) |
| 113 | \( 1 + (-8.33 + 6.60i)T \) |
| good | 5 | \( 1 + (1.31 + 2.37i)T + (-2.66 + 4.23i)T^{2} \) |
| 7 | \( 1 + (2.30 + 2.88i)T + (-1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (1.04 - 0.364i)T + (8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.295 + 2.62i)T + (-12.6 - 2.89i)T^{2} \) |
| 17 | \( 1 + (1.72 - 1.22i)T + (5.61 - 16.0i)T^{2} \) |
| 19 | \( 1 + (-0.686 + 0.197i)T + (16.0 - 10.1i)T^{2} \) |
| 23 | \( 1 + (-1.34 + 4.67i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (-1.89 - 2.66i)T + (-9.57 + 27.3i)T^{2} \) |
| 31 | \( 1 + (1.78 + 0.200i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (-2.31 + 2.07i)T + (4.14 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-3.16 - 1.10i)T + (32.0 + 25.5i)T^{2} \) |
| 43 | \( 1 + (-0.481 + 2.83i)T + (-40.5 - 14.2i)T^{2} \) |
| 47 | \( 1 + (-8.91 + 0.500i)T + (46.7 - 5.26i)T^{2} \) |
| 53 | \( 1 + (-4.83 - 10.0i)T + (-33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + (-1.29 - 4.50i)T + (-49.9 + 31.3i)T^{2} \) |
| 61 | \( 1 + (0.555 + 1.58i)T + (-47.6 + 38.0i)T^{2} \) |
| 67 | \( 1 + (-0.456 + 8.12i)T + (-66.5 - 7.50i)T^{2} \) |
| 71 | \( 1 + (-3.02 - 7.30i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-10.9 - 4.52i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-7.16 + 8.02i)T + (-8.84 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.881 - 3.86i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (11.4 - 1.94i)T + (84.0 - 29.3i)T^{2} \) |
| 97 | \( 1 + (8.51 - 10.6i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.43292754061773568647694955074, −9.425088602614583934273561811031, −8.509953001164704003036301577047, −7.62942564175861301532373038227, −6.74543687232728058014271970017, −5.48438551994983877016756557913, −4.42855136205201729372169586881, −3.85687472481158125023037107751, −2.72127124902563700652638941179, −0.793606316202284599396310932968,
2.33260683827871899200173036533, 3.07828962419992308803171587472, 3.97227528951767434537238257759, 5.42240001203395759344188206914, 6.42360510792518080268752366866, 7.03438156171529799047705957191, 7.924775400413951465513044213277, 8.993585570465604459056391292615, 9.777658409284417329408412803618, 11.00169774684163840178803277841
