| L(s) = 1 | + 2·2-s + 2·4-s + 6·7-s + 12·14-s − 4·16-s + 14·17-s − 8·23-s + 25-s + 12·28-s − 16·31-s − 8·32-s + 28·34-s − 4·41-s − 16·46-s + 14·47-s + 13·49-s + 2·50-s − 32·62-s − 8·64-s + 28·68-s + 6·71-s + 28·73-s − 20·79-s − 8·82-s − 16·92-s + 28·94-s + 16·97-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 2.26·7-s + 3.20·14-s − 16-s + 3.39·17-s − 1.66·23-s + 1/5·25-s + 2.26·28-s − 2.87·31-s − 1.41·32-s + 4.80·34-s − 0.624·41-s − 2.35·46-s + 2.04·47-s + 13/7·49-s + 0.282·50-s − 4.06·62-s − 64-s + 3.39·68-s + 0.712·71-s + 3.27·73-s − 2.25·79-s − 0.883·82-s − 1.66·92-s + 2.88·94-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.207678649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.207678649\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.54879325857465538070230369061, −9.810929708715935350486376354643, −9.568344033548428929447615449757, −9.002976634679524268556987485372, −8.256892312600616901149994718986, −8.189352302905368400678882268362, −7.74795126446030632420585194532, −7.23009806150085706545676071888, −7.04750305861168250111128132207, −5.90549497669378764643639949139, −5.80182269904285993340750573937, −5.41375683031812336635310272242, −5.12351612177070882626560888552, −4.54168481132733101164581341578, −4.06336662775372832913137138087, −3.45500436521338596356623048772, −3.30337636812120140812732157124, −2.07875543361596919789728301552, −1.91542847886084698137045959320, −1.01251428109686507359472261915,
1.01251428109686507359472261915, 1.91542847886084698137045959320, 2.07875543361596919789728301552, 3.30337636812120140812732157124, 3.45500436521338596356623048772, 4.06336662775372832913137138087, 4.54168481132733101164581341578, 5.12351612177070882626560888552, 5.41375683031812336635310272242, 5.80182269904285993340750573937, 5.90549497669378764643639949139, 7.04750305861168250111128132207, 7.23009806150085706545676071888, 7.74795126446030632420585194532, 8.189352302905368400678882268362, 8.256892312600616901149994718986, 9.002976634679524268556987485372, 9.568344033548428929447615449757, 9.810929708715935350486376354643, 10.54879325857465538070230369061
