L(s) = 1 | + (2.78 + 4.15i)5-s − 0.701i·7-s + 0.462i·11-s + 14.1i·13-s + 14.3·17-s + 5.37·19-s − 41.6·23-s + (−9.51 + 23.1i)25-s + 9.34i·29-s + 21.8·31-s + (2.91 − 1.95i)35-s + 34.0i·37-s + 22.3i·41-s − 55.9i·43-s − 7.64·47-s + ⋯ |
L(s) = 1 | + (0.556 + 0.830i)5-s − 0.100i·7-s + 0.0420i·11-s + 1.08i·13-s + 0.846·17-s + 0.283·19-s − 1.81·23-s + (−0.380 + 0.924i)25-s + 0.322i·29-s + 0.705·31-s + (0.0833 − 0.0558i)35-s + 0.921i·37-s + 0.545i·41-s − 1.30i·43-s − 0.162·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.636769838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636769838\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.78 - 4.15i)T \) |
good | 7 | \( 1 + 0.701iT - 49T^{2} \) |
| 11 | \( 1 - 0.462iT - 121T^{2} \) |
| 13 | \( 1 - 14.1iT - 169T^{2} \) |
| 17 | \( 1 - 14.3T + 289T^{2} \) |
| 19 | \( 1 - 5.37T + 361T^{2} \) |
| 23 | \( 1 + 41.6T + 529T^{2} \) |
| 29 | \( 1 - 9.34iT - 841T^{2} \) |
| 31 | \( 1 - 21.8T + 961T^{2} \) |
| 37 | \( 1 - 34.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 22.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 7.64T + 2.20e3T^{2} \) |
| 53 | \( 1 + 36.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 35.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 58.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 61.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 43.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 50.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 96.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 26.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 125. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 117. iT - 9.40e3T^{2} \) |
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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
−9.718395345526057557815931517350, −8.725553729710621796379052318711, −7.79745638655071183506005100056, −7.02719755614306068119123288467, −6.27155833631441502649930026129, −5.57391390215084944932501394486, −4.41026449069829163176924607461, −3.48699412769306306652896791402, −2.44495281877903560309517879172, −1.44846028528891028396991800848,
0.43212860782769260128660495165, 1.60109139802548805865792617985, 2.75438490784147552500721035520, 3.89210748452808537848676394125, 4.88770832890553898687605214979, 5.73389499478471142512471699124, 6.19829648721269948380921907041, 7.63077462691155090198466135634, 8.076938247212698044212633300561, 8.943330783911751130454725986499