| L(s) = 1 | + 4·5-s − 4·7-s − 4·11-s + 4·17-s + 8·23-s + 4·25-s + 4·29-s − 12·31-s − 16·35-s + 8·37-s + 12·41-s − 8·43-s + 8·47-s + 4·53-s − 16·55-s + 8·59-s + 8·61-s − 16·67-s + 24·71-s + 8·73-s + 16·77-s − 20·79-s + 4·83-s + 16·85-s − 4·89-s + 4·97-s − 4·101-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s − 1.20·11-s + 0.970·17-s + 1.66·23-s + 4/5·25-s + 0.742·29-s − 2.15·31-s − 2.70·35-s + 1.31·37-s + 1.87·41-s − 1.21·43-s + 1.16·47-s + 0.549·53-s − 2.15·55-s + 1.04·59-s + 1.02·61-s − 1.95·67-s + 2.84·71-s + 0.936·73-s + 1.82·77-s − 2.25·79-s + 0.439·83-s + 1.73·85-s − 0.423·89-s + 0.406·97-s − 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.996977842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996977842\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.375329618262859545512598849158, −8.338648029527522788207397302106, −7.58532572416580096848184173076, −7.43908389151436128273852012116, −6.92002365114007219002765867165, −6.76818470590314332507531744575, −6.05212097574748465311869877049, −6.04242688218214418044926943561, −5.62611463452801798363922552433, −5.29651257918798959001116187604, −5.06134787782277410824677783067, −4.43301874502472002572152263115, −3.89011414365806996401515786361, −3.38526823882755054085820639687, −3.06296224300242875096020734309, −2.61019775133102439087139649584, −2.30406117315868249184316057808, −1.79314128275632791846389012194, −1.04958712801019403061432655228, −0.52078042396469810305240631539,
0.52078042396469810305240631539, 1.04958712801019403061432655228, 1.79314128275632791846389012194, 2.30406117315868249184316057808, 2.61019775133102439087139649584, 3.06296224300242875096020734309, 3.38526823882755054085820639687, 3.89011414365806996401515786361, 4.43301874502472002572152263115, 5.06134787782277410824677783067, 5.29651257918798959001116187604, 5.62611463452801798363922552433, 6.04242688218214418044926943561, 6.05212097574748465311869877049, 6.76818470590314332507531744575, 6.92002365114007219002765867165, 7.43908389151436128273852012116, 7.58532572416580096848184173076, 8.338648029527522788207397302106, 8.375329618262859545512598849158
