| L(s) = 1 | − 2·3-s − 4·7-s + 3·9-s + 8·21-s − 8·23-s − 4·27-s + 8·29-s + 20·41-s + 8·43-s − 16·47-s − 2·49-s + 20·61-s − 12·63-s + 16·67-s + 16·69-s + 5·81-s + 8·83-s − 16·87-s + 12·89-s + 24·101-s + 28·103-s + 24·107-s + 20·109-s − 2·121-s − 40·123-s + 127-s − 16·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 9-s + 1.74·21-s − 1.66·23-s − 0.769·27-s + 1.48·29-s + 3.12·41-s + 1.21·43-s − 2.33·47-s − 2/7·49-s + 2.56·61-s − 1.51·63-s + 1.95·67-s + 1.92·69-s + 5/9·81-s + 0.878·83-s − 1.71·87-s + 1.27·89-s + 2.38·101-s + 2.75·103-s + 2.32·107-s + 1.91·109-s − 0.181·121-s − 3.60·123-s + 0.0887·127-s − 1.40·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.295606870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295606870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.394923561499792875278883090299, −8.846023156417931987813042911767, −8.268163461270784838807758700787, −8.054586098693156195597943569590, −7.37472326580015389290904311162, −7.33687140914472755794897013319, −6.55678835939007687340089477271, −6.37636675435742967614936655791, −6.07389078485260699572800093646, −5.95339759999321565514956373542, −5.07083261314643721631974564941, −5.02709985439448911595611073866, −4.22195476922834230565300207833, −4.11514933551697702845382619408, −3.32972740334086786409784125902, −3.18689732587624555397119038860, −2.19614766026055845183044342987, −2.08109084610112438217265453635, −0.73889573525456845550463854024, −0.64799272671696945723453389666,
0.64799272671696945723453389666, 0.73889573525456845550463854024, 2.08109084610112438217265453635, 2.19614766026055845183044342987, 3.18689732587624555397119038860, 3.32972740334086786409784125902, 4.11514933551697702845382619408, 4.22195476922834230565300207833, 5.02709985439448911595611073866, 5.07083261314643721631974564941, 5.95339759999321565514956373542, 6.07389078485260699572800093646, 6.37636675435742967614936655791, 6.55678835939007687340089477271, 7.33687140914472755794897013319, 7.37472326580015389290904311162, 8.054586098693156195597943569590, 8.268163461270784838807758700787, 8.846023156417931987813042911767, 9.394923561499792875278883090299
