L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.239 + 1.66i)5-s + (−0.841 − 0.540i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.698 − 1.53i)10-s + (−0.797 − 0.234i)11-s + (0.959 + 0.281i)12-s + (0.142 − 0.989i)14-s + (−1.10 + 1.27i)15-s + (0.415 − 0.909i)16-s + (1.10 + 0.708i)17-s + (−0.142 − 0.989i)18-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.239 + 1.66i)5-s + (−0.841 − 0.540i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.698 − 1.53i)10-s + (−0.797 − 0.234i)11-s + (0.959 + 0.281i)12-s + (0.142 − 0.989i)14-s + (−1.10 + 1.27i)15-s + (0.415 − 0.909i)16-s + (1.10 + 0.708i)17-s + (−0.142 − 0.989i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9172216638\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9172216638\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-0.841 + 0.540i)T \) |
good | 5 | \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 17 | \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 31 | \( 1 + (0.857 - 0.989i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 41 | \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 67 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (0.186 + 0.215i)T + (-0.142 + 0.989i)T^{2} \) |
| 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
−9.652637016740075895168916256794, −9.070327401580637590695716623102, −8.258212911714623034759264359570, −7.37983191009325406285645874369, −6.89335154445240572972099843148, −5.70228367680766656135390485725, −5.30202094600210012297800601103, −3.28740016554311250783842694068, −3.00349458720770160858855282978, −2.11159073183206556254884432150,
0.901852498518305055044524895373, 1.51372495642531462859557610760, 2.93231438604335741825163997380, 3.73697843091771221004104303347, 5.09588364385342948452694286386, 5.96712416123406714444967974386, 7.17575043400845565617441213673, 7.77888301559227863069618261065, 8.093368000702737466415280710771, 9.219208546391543600129286551375