L(s) = 1 | + (0.809 − 0.587i)3-s + (−0.587 − 1.80i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (−1.53 − 1.11i)15-s + (0.363 + 1.11i)17-s + (1.30 − 0.951i)19-s − 21-s − 1.17·23-s + (−2.11 + 1.53i)25-s + (−0.309 − 0.951i)27-s + (−0.190 + 0.587i)31-s + (−0.951 + 0.309i)33-s + (−0.587 + 1.80i)35-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)3-s + (−0.587 − 1.80i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (−1.53 − 1.11i)15-s + (0.363 + 1.11i)17-s + (1.30 − 0.951i)19-s − 21-s − 1.17·23-s + (−2.11 + 1.53i)25-s + (−0.309 − 0.951i)27-s + (−0.190 + 0.587i)31-s + (−0.951 + 0.309i)33-s + (−0.587 + 1.80i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075126554\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075126554\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.951 + 0.309i)T \) |
good | 5 | \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + 1.17T + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 1.17T + T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−8.268061988298508502009292261088, −7.75768861475043386457022687972, −7.19566272909059935646616870099, −6.03583249497449403997268471201, −5.36101765207498669829490868611, −4.31014400734032095180165396701, −3.73465920113198479072491902616, −2.81096374143123245887085625999, −1.50061263828955920679368889993, −0.55210809854561069690562837197,
2.33081008907206911532924080379, 2.81314135606133321521601286145, 3.43243073441645266878428113684, 4.18368478002953983523449995656, 5.40692549224645970613982802396, 6.10126166023432475640452591824, 7.08539997163606391229571044186, 7.75090318884627669509567094069, 7.971766567316372533240957566924, 9.331550516843396145517753814225