L(s) = 1 | − 4·5-s − 2·7-s + 5·11-s − 2·13-s + 3·17-s + 19-s + 6·23-s + 11·25-s + 2·29-s − 4·31-s + 8·35-s − 8·37-s − 41-s − 7·43-s − 2·47-s − 3·49-s + 4·53-s − 20·55-s − 5·59-s + 8·65-s − 13·67-s − 8·71-s + 3·73-s − 10·77-s + 8·79-s − 12·83-s − 12·85-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 0.755·7-s + 1.50·11-s − 0.554·13-s + 0.727·17-s + 0.229·19-s + 1.25·23-s + 11/5·25-s + 0.371·29-s − 0.718·31-s + 1.35·35-s − 1.31·37-s − 0.156·41-s − 1.06·43-s − 0.291·47-s − 3/7·49-s + 0.549·53-s − 2.69·55-s − 0.650·59-s + 0.992·65-s − 1.58·67-s − 0.949·71-s + 0.351·73-s − 1.13·77-s + 0.900·79-s − 1.31·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.571372153079851950591634360901, −7.60587858922252495814577026225, −7.05091284851179381590000850217, −6.47625713727980907317494014975, −5.21398253698213513691070232995, −4.37221273382751064149953009765, −3.52729988182521290445317913211, −3.11552562246682735877624015370, −1.28502141448365930031973361559, 0,
1.28502141448365930031973361559, 3.11552562246682735877624015370, 3.52729988182521290445317913211, 4.37221273382751064149953009765, 5.21398253698213513691070232995, 6.47625713727980907317494014975, 7.05091284851179381590000850217, 7.60587858922252495814577026225, 8.571372153079851950591634360901