| L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 13-s − 16-s − 3·17-s + 26-s + 29-s + 3·34-s − 37-s + 40-s + 3·41-s − 49-s − 58-s − 61-s + 64-s − 65-s + 2·73-s + 74-s − 80-s − 3·82-s − 3·85-s − 2·97-s + 98-s − 101-s − 104-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 13-s − 16-s − 3·17-s + 26-s + 29-s + 3·34-s − 37-s + 40-s + 3·41-s − 49-s − 58-s − 61-s + 64-s − 65-s + 2·73-s + 74-s − 80-s − 3·82-s − 3·85-s − 2·97-s + 98-s − 101-s − 104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5658311754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5658311754\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.451415212167278172449171789783, −8.972337002872417002452170565700, −8.738941703944119702979204921167, −8.401778810997543046604790358185, −7.87082337847413851745786429646, −7.51601375782960703317785784678, −7.11874892192957121085948739382, −6.69019193514015482135868879667, −6.36594065584998593666017584046, −6.05369756595507505462621147984, −5.40167907940970704936161584671, −4.82929680043790461930194013427, −4.78968537723391802193478270294, −4.12843145676526470424961264557, −3.92012694808935278542603996351, −2.73191806492029992542916402400, −2.63513941741403956007444998458, −1.96000304757661680868619476886, −1.67726933105061520128119459838, −0.59399449254363931391372537060,
0.59399449254363931391372537060, 1.67726933105061520128119459838, 1.96000304757661680868619476886, 2.63513941741403956007444998458, 2.73191806492029992542916402400, 3.92012694808935278542603996351, 4.12843145676526470424961264557, 4.78968537723391802193478270294, 4.82929680043790461930194013427, 5.40167907940970704936161584671, 6.05369756595507505462621147984, 6.36594065584998593666017584046, 6.69019193514015482135868879667, 7.11874892192957121085948739382, 7.51601375782960703317785784678, 7.87082337847413851745786429646, 8.401778810997543046604790358185, 8.738941703944119702979204921167, 8.972337002872417002452170565700, 9.451415212167278172449171789783
