L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s − 0.999·12-s + (−0.5 − 0.866i)16-s + 17-s + 0.999·18-s + 19-s + (0.499 − 0.866i)22-s + (0.5 + 0.866i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s − 0.999·12-s + (−0.5 − 0.866i)16-s + 17-s + 0.999·18-s + 19-s + (0.499 − 0.866i)22-s + (0.5 + 0.866i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.148753337\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.148753337\) |
\(L(1)\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.142792477057806123827918849226, −8.114783648399841316673735849006, −7.81293167792934109801927586975, −6.82424923555793465258863066796, −5.56120935146871348339114457007, −4.72125741956824816919274408551, −4.04260115477821288844358963984, −3.26256151468185372240320607177, −2.47221110691323698640157847026, −1.37474811863420541552851100091,
0.853757497419021379698111168003, 1.78387239858480799404086600067, 3.16202888327544439705383499544, 3.91513948962730953183343170928, 5.36029001331115272600287279806, 5.74777021942009769818086655632, 6.65082042375633154700854803211, 7.26958983436783143438935801586, 7.896955186343717929773410806403, 8.486533330801220667603427185049