L(s) = 1 | + (0.984 + 0.173i)2-s + (0.223 − 0.266i)3-s + (0.939 + 0.342i)4-s + (0.266 − 0.223i)6-s + (0.866 + 0.5i)8-s + (0.152 + 0.866i)9-s + (−0.766 + 1.32i)11-s + (0.300 − 0.173i)12-s + (0.766 + 0.642i)16-s + (−0.984 − 0.173i)17-s + 0.879i·18-s + (0.939 + 0.342i)19-s + (−0.984 + 1.17i)22-s + (0.326 − 0.118i)24-s + (0.565 + 0.326i)27-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)2-s + (0.223 − 0.266i)3-s + (0.939 + 0.342i)4-s + (0.266 − 0.223i)6-s + (0.866 + 0.5i)8-s + (0.152 + 0.866i)9-s + (−0.766 + 1.32i)11-s + (0.300 − 0.173i)12-s + (0.766 + 0.642i)16-s + (−0.984 − 0.173i)17-s + 0.879i·18-s + (0.939 + 0.342i)19-s + (−0.984 + 1.17i)22-s + (0.326 − 0.118i)24-s + (0.565 + 0.326i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.564591068\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.564591068\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.939 - 0.342i)T \) |
good | 3 | \( 1 + (-0.223 + 0.266i)T + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.984 + 0.173i)T + (0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (-1.85 + 0.326i)T + (0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.984 + 1.17i)T + (-0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (1.85 + 0.326i)T + (0.939 + 0.342i)T^{2} \) |
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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
−8.497638138915597456868098015697, −7.82438953595717737330595674044, −7.14734488412870736125839430623, −6.78752464941552185460477317463, −5.52254067623321079358164152348, −5.05870067220500861972040943591, −4.35793762837041132664249679718, −3.37737966020233116228054691468, −2.31848689503835565338050141733, −1.86142792454696920686241921598,
1.04733486067207480413648154690, 2.46660746514267459673998725846, 3.22630641950001071370301014637, 3.80527691025586793763544819722, 4.81726183582903685529826011209, 5.42140509182043559779380596361, 6.37476315176169867838479997077, 6.73696974212283378637355791053, 7.86206919620370777490836647512, 8.461525314587771686089844234142