L(s) = 1 | + 2·3-s + 7-s + 9-s + 4·13-s + 6·19-s + 2·21-s + 3·23-s − 4·27-s + 3·29-s − 9·37-s + 8·39-s − 2·41-s − 9·43-s − 6·47-s + 49-s − 6·53-s + 12·57-s − 8·59-s + 10·61-s + 63-s − 67-s + 6·69-s + 7·71-s − 2·73-s − 9·79-s − 11·81-s − 12·83-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s + 1.37·19-s + 0.436·21-s + 0.625·23-s − 0.769·27-s + 0.557·29-s − 1.47·37-s + 1.28·39-s − 0.312·41-s − 1.37·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 1.58·57-s − 1.04·59-s + 1.28·61-s + 0.125·63-s − 0.122·67-s + 0.722·69-s + 0.830·71-s − 0.234·73-s − 1.01·79-s − 1.22·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.95478062044176, −12.39334848833419, −11.83934692357647, −11.35913847186641, −11.16670774967720, −10.38908048368797, −10.08729586307051, −9.384521726864157, −9.233839245963296, −8.500086535550397, −8.291456630360295, −8.018962552082219, −7.220044249970423, −6.954716842254102, −6.396327034262254, −5.634863208688260, −5.352328124996653, −4.737685381832870, −4.150044254251995, −3.514881692317238, −3.141607523272132, −2.884647997960509, −1.919319109630624, −1.580901691557802, −0.9834234712871822, 0,
0.9834234712871822, 1.580901691557802, 1.919319109630624, 2.884647997960509, 3.141607523272132, 3.514881692317238, 4.150044254251995, 4.737685381832870, 5.352328124996653, 5.634863208688260, 6.396327034262254, 6.954716842254102, 7.220044249970423, 8.018962552082219, 8.291456630360295, 8.500086535550397, 9.233839245963296, 9.384521726864157, 10.08729586307051, 10.38908048368797, 11.16670774967720, 11.35913847186641, 11.83934692357647, 12.39334848833419, 12.95478062044176