L(s) = 1 | + (−0.5 + 0.866i)3-s + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 1.73i·21-s + (−1.5 − 0.866i)23-s + 0.999·27-s + (−0.5 − 0.866i)29-s + (−0.5 + 0.866i)41-s + (1.5 − 0.866i)47-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (1.49 + 0.866i)63-s + (−1.5 − 0.866i)67-s + (1.5 − 0.866i)69-s + (−0.5 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 1.73i·21-s + (−1.5 − 0.866i)23-s + 0.999·27-s + (−0.5 − 0.866i)29-s + (−0.5 + 0.866i)41-s + (1.5 − 0.866i)47-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (1.49 + 0.866i)63-s + (−1.5 − 0.866i)67-s + (1.5 − 0.866i)69-s + (−0.5 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2727113324\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2727113324\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 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
−8.830543883062758526579946750916, −7.950821820276303306155377821785, −6.80518579417157308514338345250, −6.09038764979851478767533676633, −5.80180359799048099296536788634, −4.73977578504509466980451232513, −3.89103169893303500081729982636, −3.16050658357450261029861473460, −2.24532460033465538263627394744, −0.17670312809240730142134776671,
1.20214410967519935251523772018, 2.41442422883984642557358974120, 3.43693181128580221651685283601, 4.16922672851381335766982729892, 5.40602217646731782991194303894, 6.04059374890192725873992573803, 6.69321995569481288978389517780, 7.35014736276083701131312028986, 7.84801008363821895872030989119, 8.950471305527200165913420314449