L(s) = 1 | + (−2 + 3.46i)5-s + 3·7-s + 2·11-s + (3.5 + 6.06i)13-s + (4 − 1.73i)19-s + (2 + 3.46i)23-s + (−5.49 − 9.52i)25-s + (2 + 3.46i)29-s − 31-s + (−6 + 10.3i)35-s + 7·37-s + (2 − 3.46i)41-s + (3.5 − 6.06i)43-s + (−1 − 1.73i)47-s + 2·49-s + ⋯ |
L(s) = 1 | + (−0.894 + 1.54i)5-s + 1.13·7-s + 0.603·11-s + (0.970 + 1.68i)13-s + (0.917 − 0.397i)19-s + (0.417 + 0.722i)23-s + (−1.09 − 1.90i)25-s + (0.371 + 0.643i)29-s − 0.179·31-s + (−1.01 + 1.75i)35-s + 1.15·37-s + (0.312 − 0.541i)41-s + (0.533 − 0.924i)43-s + (−0.145 − 0.252i)47-s + 0.285·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.999629744\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.999629744\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + (-2 + 3.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.964352994183173528559712572281, −8.173944345849538357623051596160, −7.36535517180210490083423337798, −6.90768362418068808129872231207, −6.19503066982407263835650117920, −5.03398695111569728729471972275, −4.04678939340508269137047507859, −3.57165129799234370174013546857, −2.42247961769237520801921498526, −1.32468323427010767689153182458,
0.814222771209684650277867962788, 1.33301844048870895814082791917, 3.00675521656240391288960224568, 4.05957846440704100496657405348, 4.63727952274961801821900700077, 5.40237620230378471237568583622, 6.09581380973770532665644574178, 7.52995208028942778429812674112, 8.018101391684494754588556326015, 8.425846009219178909549921596084