L(s) = 1 | + (0.623 − 0.781i)2-s + (1.62 + 0.781i)3-s + (−0.222 − 0.974i)4-s + (1.62 − 0.781i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (1.40 + 1.75i)9-s + (0.777 − 0.974i)11-s + (0.400 − 1.75i)12-s + (0.777 − 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + 2.24·18-s + (−1.12 − 1.40i)21-s + (−0.277 − 1.21i)22-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)2-s + (1.62 + 0.781i)3-s + (−0.222 − 0.974i)4-s + (1.62 − 0.781i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (1.40 + 1.75i)9-s + (0.777 − 0.974i)11-s + (0.400 − 1.75i)12-s + (0.777 − 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + 2.24·18-s + (−1.12 − 1.40i)21-s + (−0.277 − 1.21i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.794472636\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.794472636\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.623 + 0.781i)T \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
| 17 | \( 1 + (0.222 - 0.974i)T \) |
good | 3 | \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 5 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + 1.80T + T^{2} \) |
| 37 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 61 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.816847783384506408862439114889, −8.385251299017871601771258033438, −7.24447633704381236107552869307, −6.26884828633074220788346536417, −5.53402952251152962625524970097, −4.29502192060727304995159696964, −3.73745350884794681555926611202, −3.30978354497273307790501486054, −2.54755603560408072253024734036, −1.29223192304533435226780931431,
1.78167761231273215605079634324, 2.57563752139687124372397925501, 3.59836164524781691976066744660, 3.91918708090996316887330666189, 5.12526868749170219000549225796, 6.33066190217200478284695899443, 6.81375933264209262423243254519, 7.27780830565181572790483934944, 8.073406986954312727797597961172, 8.969187130111730971453306582358