| L(s) = 1 | + 4-s − 4·7-s − 2·13-s + 16-s − 12·19-s − 6·25-s − 4·28-s − 4·31-s + 12·37-s − 16·43-s − 2·49-s − 2·52-s + 20·61-s + 64-s − 4·67-s + 28·73-s − 12·76-s − 8·79-s + 8·91-s − 20·97-s − 6·100-s + 8·103-s + 20·109-s − 4·112-s − 6·121-s − 4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.51·7-s − 0.554·13-s + 1/4·16-s − 2.75·19-s − 6/5·25-s − 0.755·28-s − 0.718·31-s + 1.97·37-s − 2.43·43-s − 2/7·49-s − 0.277·52-s + 2.56·61-s + 1/8·64-s − 0.488·67-s + 3.27·73-s − 1.37·76-s − 0.900·79-s + 0.838·91-s − 2.03·97-s − 3/5·100-s + 0.788·103-s + 1.91·109-s − 0.377·112-s − 0.545·121-s − 0.359·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.773699703719295926726447008045, −9.511260567162705652960388658514, −8.695739660347841822203445826112, −8.241527102852606326983085030222, −7.76516545555292037952264969735, −6.93707785293490008685942511031, −6.49638165287752346862203973768, −6.33856513547641234727266817168, −5.59584593532556316161391417712, −4.80645914681781210957697738257, −3.99394475360161838020150844401, −3.53791924823882088462258917674, −2.57523735650192716532994366231, −2.01663812390586556648231292076, 0,
2.01663812390586556648231292076, 2.57523735650192716532994366231, 3.53791924823882088462258917674, 3.99394475360161838020150844401, 4.80645914681781210957697738257, 5.59584593532556316161391417712, 6.33856513547641234727266817168, 6.49638165287752346862203973768, 6.93707785293490008685942511031, 7.76516545555292037952264969735, 8.241527102852606326983085030222, 8.695739660347841822203445826112, 9.511260567162705652960388658514, 9.773699703719295926726447008045
