L(s) = 1 | + (0.766 − 0.642i)2-s + (0.939 − 0.342i)5-s + (0.499 + 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.173 + 0.984i)31-s + (0.173 + 0.984i)34-s + (0.939 + 0.342i)38-s + (0.766 + 0.642i)40-s + (−0.5 − 0.866i)46-s + (−0.347 − 1.96i)47-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.939 − 0.342i)5-s + (0.499 + 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.173 + 0.984i)31-s + (0.173 + 0.984i)34-s + (0.939 + 0.342i)38-s + (0.766 + 0.642i)40-s + (−0.5 − 0.866i)46-s + (−0.347 − 1.96i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.318611058\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.318611058\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
good | 2 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 11 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (0.347 + 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
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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
−8.553401538208625434946698925066, −8.231090192139207459638002740463, −7.07245752228965220560696685070, −6.27165516265501317857536678103, −5.46161380767578513949012951304, −4.86164297350243892168872281953, −4.01776355698430683460293546805, −3.20001487732255687435525563163, −2.24060517984886462858065627985, −1.51221687213511594313654647007,
1.21456077561508205553633504705, 2.43463959472793624754975509714, 3.33892953620750641831089407186, 4.44087938284334154727833098831, 5.10701314956151739076937464585, 5.75282250613648301249686489316, 6.39965827385345759188077257911, 7.11890118926210149991818260096, 7.61891776585146214870432152337, 8.893216299360211226106379982176