| L(s) = 1 | − 2-s + 4-s + 4·5-s + 4·7-s − 8-s − 2·9-s − 4·10-s + 4·11-s − 10·13-s − 4·14-s + 16-s + 2·18-s + 4·20-s − 4·22-s + 11·25-s + 10·26-s + 4·28-s + 16·31-s − 32-s + 16·35-s − 2·36-s − 4·40-s + 4·44-s − 8·45-s + 9·49-s − 11·50-s − 10·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 1.51·7-s − 0.353·8-s − 2/3·9-s − 1.26·10-s + 1.20·11-s − 2.77·13-s − 1.06·14-s + 1/4·16-s + 0.471·18-s + 0.894·20-s − 0.852·22-s + 11/5·25-s + 1.96·26-s + 0.755·28-s + 2.87·31-s − 0.176·32-s + 2.70·35-s − 1/3·36-s − 0.632·40-s + 0.603·44-s − 1.19·45-s + 9/7·49-s − 1.55·50-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.878058677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878058677\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−9.440125352225763426371104427070, −8.758134343767079211035259473666, −8.452851881947536914606955446437, −7.88354468708042569174650283422, −7.33273399204943944544007458185, −6.83808481106766006891219698909, −6.35588369309814810989538386386, −5.80854892500226407155934273417, −5.20851867257799630766010860838, −4.79144239074668791235485504609, −4.36605133941615168008745513303, −2.93285326926119634938672931622, −2.47648340869567352289993240339, −1.94053351573316158541227551736, −1.11188998389765416864702733305,
1.11188998389765416864702733305, 1.94053351573316158541227551736, 2.47648340869567352289993240339, 2.93285326926119634938672931622, 4.36605133941615168008745513303, 4.79144239074668791235485504609, 5.20851867257799630766010860838, 5.80854892500226407155934273417, 6.35588369309814810989538386386, 6.83808481106766006891219698909, 7.33273399204943944544007458185, 7.88354468708042569174650283422, 8.452851881947536914606955446437, 8.758134343767079211035259473666, 9.440125352225763426371104427070
