| L(s) = 1 | − 8·3-s + 5·5-s − 2·7-s + 37·9-s − 22·11-s + 10·13-s − 40·15-s + 10·17-s − 110·19-s + 16·21-s + 154·23-s + 25·25-s − 80·27-s + 222·29-s + 92·31-s + 176·33-s − 10·35-s − 34·37-s − 80·39-s + 398·41-s − 268·43-s + 185·45-s + 10·47-s − 339·49-s − 80·51-s − 582·53-s − 110·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.447·5-s − 0.107·7-s + 1.37·9-s − 0.603·11-s + 0.213·13-s − 0.688·15-s + 0.142·17-s − 1.32·19-s + 0.166·21-s + 1.39·23-s + 1/5·25-s − 0.570·27-s + 1.42·29-s + 0.533·31-s + 0.928·33-s − 0.0482·35-s − 0.151·37-s − 0.328·39-s + 1.51·41-s − 0.950·43-s + 0.612·45-s + 0.0310·47-s − 0.988·49-s − 0.219·51-s − 1.50·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 10 T + p^{3} T^{2} \) |
| 19 | \( 1 + 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 154 T + p^{3} T^{2} \) |
| 29 | \( 1 - 222 T + p^{3} T^{2} \) |
| 31 | \( 1 - 92 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 398 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 10 T + p^{3} T^{2} \) |
| 53 | \( 1 + 582 T + p^{3} T^{2} \) |
| 59 | \( 1 + 746 T + p^{3} T^{2} \) |
| 61 | \( 1 + 226 T + p^{3} T^{2} \) |
| 67 | \( 1 - 172 T + p^{3} T^{2} \) |
| 71 | \( 1 + 928 T + p^{3} T^{2} \) |
| 73 | \( 1 - 570 T + p^{3} T^{2} \) |
| 79 | \( 1 + 64 T + p^{3} T^{2} \) |
| 83 | \( 1 - 864 T + p^{3} T^{2} \) |
| 89 | \( 1 + 874 T + p^{3} T^{2} \) |
| 97 | \( 1 - 306 T + p^{3} T^{2} \) |
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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
−10.05610527338700054369181767966, −8.982791372923607952652750353898, −7.88197632771061044933091515375, −6.61603760641222338062586427192, −6.22854821619418556197304753043, −5.15544073209096385890669609667, −4.51996851136921380719994965930, −2.84286162889573564858687314714, −1.26574578594714129764064180760, 0,
1.26574578594714129764064180760, 2.84286162889573564858687314714, 4.51996851136921380719994965930, 5.15544073209096385890669609667, 6.22854821619418556197304753043, 6.61603760641222338062586427192, 7.88197632771061044933091515375, 8.982791372923607952652750353898, 10.05610527338700054369181767966
