L(s) = 1 | + (0.900 + 0.433i)3-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)12-s + (0.900 − 0.433i)13-s + (−0.900 + 0.433i)16-s − 1.56i·19-s + (0.900 − 0.433i)21-s + (−0.623 − 0.781i)25-s + (0.222 + 0.974i)27-s + (0.900 + 0.433i)28-s + 1.94i·31-s + (−0.623 + 0.781i)36-s + (−1.90 − 0.433i)37-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)3-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)12-s + (0.900 − 0.433i)13-s + (−0.900 + 0.433i)16-s − 1.56i·19-s + (0.900 − 0.433i)21-s + (−0.623 − 0.781i)25-s + (0.222 + 0.974i)27-s + (0.900 + 0.433i)28-s + 1.94i·31-s + (−0.623 + 0.781i)36-s + (−1.90 − 0.433i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.776564078\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.776564078\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.900 - 0.433i)T \) |
| 7 | \( 1 + (-0.623 + 0.781i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 5 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 + 1.56iT - T^{2} \) |
| 23 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 - 1.94iT - T^{2} \) |
| 37 | \( 1 + (1.90 + 0.433i)T + (0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \) |
| 67 | \( 1 + 0.867iT - T^{2} \) |
| 71 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (1.52 - 1.21i)T + (0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 - 0.445T + T^{2} \) |
| 83 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 - 1.56iT - T^{2} \) |
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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
−9.255037275817857160204173756070, −8.491887583732759687228973183606, −8.191439363886392289506302363580, −7.20226066353901857683397735999, −6.74863419113681012677248369629, −5.16248065510141547902628701843, −4.39114245307915029501003394315, −3.57970090789448985897369888224, −2.88686341112339342896614129455, −1.67625767291194463698634079336,
1.57880841193952383045180349575, 1.98204630619652994907345182472, 3.33491112302346364596449495966, 4.29829218620317679489311199556, 5.51733308008598699959458055998, 6.05126293125960774214609657160, 6.94462137049488693397770526869, 7.88558131089529886729746751958, 8.523305555326470349092224662548, 9.249978974186229296489548301581