L(s) = 1 | + (0.368 + 0.425i)3-s + (0.142 + 0.989i)5-s + (−0.755 + 1.65i)7-s + (0.0971 − 0.675i)9-s + (−0.368 + 0.425i)15-s + (−0.983 + 0.288i)21-s + (0.281 + 0.959i)23-s + (−0.959 + 0.281i)25-s + (0.797 − 0.512i)27-s + (−0.239 − 0.153i)29-s + (−1.74 − 0.512i)35-s + (0.118 + 0.822i)41-s + (0.708 + 0.817i)43-s + 0.682·45-s − 1.97·47-s + ⋯ |
L(s) = 1 | + (0.368 + 0.425i)3-s + (0.142 + 0.989i)5-s + (−0.755 + 1.65i)7-s + (0.0971 − 0.675i)9-s + (−0.368 + 0.425i)15-s + (−0.983 + 0.288i)21-s + (0.281 + 0.959i)23-s + (−0.959 + 0.281i)25-s + (0.797 − 0.512i)27-s + (−0.239 − 0.153i)29-s + (−1.74 − 0.512i)35-s + (0.118 + 0.822i)41-s + (0.708 + 0.817i)43-s + 0.682·45-s − 1.97·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.134136178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.134136178\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.281 - 0.959i)T \) |
good | 3 | \( 1 + (-0.368 - 0.425i)T + (-0.142 + 0.989i)T^{2} \) |
| 7 | \( 1 + (0.755 - 1.65i)T + (-0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 17 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 31 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 37 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 41 | \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (-0.708 - 0.817i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + 1.97T + T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 67 | \( 1 + (1.45 - 0.425i)T + (0.841 - 0.540i)T^{2} \) |
| 71 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (-0.215 + 1.49i)T + (-0.959 - 0.281i)T^{2} \) |
| 89 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.959 - 0.281i)T^{2} \) |
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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
−9.622884724189096992273836417323, −9.113616421640308653336459627588, −8.281654823437477102932476162243, −7.25881208343963790104250460147, −6.27895850458588596782871492747, −5.97409383065353476489264334566, −4.82339160126138421484413947235, −3.43767824944645185418180576849, −3.08390670842130696582390605450, −2.04917417356493426395222551226,
0.809994956083620348411145042731, 2.00114333289604169111408416998, 3.32037644538204256481626619195, 4.27450448825052414771172450144, 4.94852111886658032498214042375, 6.09777287273895059006414257137, 7.03999022991823787734888053762, 7.55512420670341198441025136882, 8.389757742583893545147265495466, 9.095213812156533493276482946064