L(s) = 1 | − i·2-s − 4-s + i·8-s + 16-s + (−1 + i)17-s + (−1 + i)19-s + (−1 + i)23-s − i·32-s + (1 + i)34-s + (1 + i)38-s + (1 + i)46-s + (−1 + i)47-s − i·49-s − 2·53-s + (1 + i)61-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + i·8-s + 16-s + (−1 + i)17-s + (−1 + i)19-s + (−1 + i)23-s − i·32-s + (1 + i)34-s + (1 + i)38-s + (1 + i)46-s + (−1 + i)47-s − i·49-s − 2·53-s + (1 + i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5684665075\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5684665075\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (1 - i)T - iT^{2} \) |
| 53 | \( 1 + 2T + T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + 2iT - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−8.840329471585470289009577905146, −8.307954698974186515567994325747, −7.64546647484322424397325663200, −6.40424146961632750196400319984, −5.83733642954096286456462120135, −4.80637539491969613091654765738, −4.04885870075158876883996776906, −3.43220991952788035647079883045, −2.21963929158248161733974633278, −1.56324844890515915473163182591,
0.32011811031841551413749580584, 2.09847707073508280553719848146, 3.21988968857554125625413749981, 4.43682765824532894580321547985, 4.69870004726746144078376948285, 5.75072337935010036189406773612, 6.61661409666853980558975589704, 6.88925762661635178074584972853, 7.950447178398915132708878322573, 8.464483206693136692419725031041