L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 0.999·8-s − 0.999·10-s + (−0.5 + 0.866i)11-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 2·17-s − 19-s + (−0.499 + 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + 0.999·28-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 0.999·8-s − 0.999·10-s + (−0.5 + 0.866i)11-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 2·17-s − 19-s + (−0.499 + 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1056319361\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1056319361\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 2T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.683853055733140379931782060656, −8.407984780705646861222181187635, −6.86273479009930777110412205152, −6.26219586184096792929289563233, −5.10990659886710935849476976855, −4.63512055522345209785324507288, −3.86801365133596540582806283427, −2.56828873694785192864103702697, −1.95770923021089394263121005602, −0.05589247536055822244862706505,
2.48767257372128107664000793652, 3.46958458099464259555162702871, 4.05560100228629968758842280749, 4.95591686772871293700804988672, 6.16742380293145663754479694015, 6.58636044361901304732319538454, 7.27914891303220673270311880185, 8.021798460445972395354784225105, 8.709925750952979554698179846700, 9.611528101627182073541016726324