| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 9-s + 2·10-s − 2·13-s + 16-s − 4·17-s − 18-s − 2·20-s + 3·25-s + 2·26-s + 4·29-s − 32-s + 4·34-s + 36-s + 4·37-s + 2·40-s − 12·41-s − 2·45-s + 2·49-s − 3·50-s − 2·52-s + 20·53-s − 4·58-s − 20·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.554·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.447·20-s + 3/5·25-s + 0.392·26-s + 0.742·29-s − 0.176·32-s + 0.685·34-s + 1/6·36-s + 0.657·37-s + 0.316·40-s − 1.87·41-s − 0.298·45-s + 2/7·49-s − 0.424·50-s − 0.277·52-s + 2.74·53-s − 0.525·58-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7297265440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7297265440\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.006839463932712238130939236098, −7.61499574063730421950670590635, −7.15844162242666446745691958741, −6.88165838632433844511443205543, −6.51726694243101001979545867511, −5.84834039375614857057682337615, −5.42080963650772912060550607812, −4.75089895493111371395700607457, −4.33965496590448459865286945078, −4.01177849501872584924243319427, −3.16914625298580597606741543707, −2.83072987113389395332805085297, −2.10513111563423464681881247262, −1.40608557557707214069713458551, −0.43866809856586112619492267991,
0.43866809856586112619492267991, 1.40608557557707214069713458551, 2.10513111563423464681881247262, 2.83072987113389395332805085297, 3.16914625298580597606741543707, 4.01177849501872584924243319427, 4.33965496590448459865286945078, 4.75089895493111371395700607457, 5.42080963650772912060550607812, 5.84834039375614857057682337615, 6.51726694243101001979545867511, 6.88165838632433844511443205543, 7.15844162242666446745691958741, 7.61499574063730421950670590635, 8.006839463932712238130939236098
