| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s − 2·9-s + 2·10-s + 4·13-s + 16-s + 12·17-s − 2·18-s + 2·20-s + 3·25-s + 4·26-s + 12·29-s + 32-s + 12·34-s − 2·36-s + 2·37-s + 2·40-s − 12·41-s − 4·45-s − 10·49-s + 3·50-s + 4·52-s + 12·53-s + 12·58-s − 20·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 2/3·9-s + 0.632·10-s + 1.10·13-s + 1/4·16-s + 2.91·17-s − 0.471·18-s + 0.447·20-s + 3/5·25-s + 0.784·26-s + 2.22·29-s + 0.176·32-s + 2.05·34-s − 1/3·36-s + 0.328·37-s + 0.316·40-s − 1.87·41-s − 0.596·45-s − 1.42·49-s + 0.424·50-s + 0.554·52-s + 1.64·53-s + 1.57·58-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.969683988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.969683988\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.191103303293285704368637540717, −7.38307853599763365662944352653, −7.37364372891229977185623115606, −6.40980423427948262053439128925, −6.24527726803627967890931657857, −5.87742893251240658386511463008, −5.49324095653367495167136231203, −4.80762486474812590647702092154, −4.78046421920466718576600337385, −3.69764756556049564109121180323, −3.29880100176274410685427430205, −3.06956943298945243015211313058, −2.32883355551645860271045039470, −1.44505436450462837187908934432, −1.03875136195910076855341646601,
1.03875136195910076855341646601, 1.44505436450462837187908934432, 2.32883355551645860271045039470, 3.06956943298945243015211313058, 3.29880100176274410685427430205, 3.69764756556049564109121180323, 4.78046421920466718576600337385, 4.80762486474812590647702092154, 5.49324095653367495167136231203, 5.87742893251240658386511463008, 6.24527726803627967890931657857, 6.40980423427948262053439128925, 7.37364372891229977185623115606, 7.38307853599763365662944352653, 8.191103303293285704368637540717
