| L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s − 2·9-s + 2·14-s − 16-s − 8·17-s − 2·18-s + 8·23-s − 2·25-s − 2·28-s − 6·31-s + 5·32-s − 8·34-s + 2·36-s + 8·41-s + 8·46-s + 11·47-s − 10·49-s − 2·50-s − 6·56-s − 6·62-s − 4·63-s + 7·64-s + 8·68-s + 14·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s − 2/3·9-s + 0.534·14-s − 1/4·16-s − 1.94·17-s − 0.471·18-s + 1.66·23-s − 2/5·25-s − 0.377·28-s − 1.07·31-s + 0.883·32-s − 1.37·34-s + 1/3·36-s + 1.24·41-s + 1.17·46-s + 1.60·47-s − 1.42·49-s − 0.282·50-s − 0.801·56-s − 0.762·62-s − 0.503·63-s + 7/8·64-s + 0.970·68-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8618514072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8618514072\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.94816353122832778160077816830, −12.40808583992367980459529575016, −11.58015978794591132342486769382, −11.09594181253137038874945479137, −10.76791685486241812982375068608, −9.542573495328596472916983985115, −9.037745079856304144672572411317, −8.626116878885754625836803452995, −7.80529576046407797508604643325, −6.89452086470496642181392521129, −6.09945175507229631225492548529, −5.24479535634694968816093872884, −4.67640780536884506239351662981, −3.78664716997013961926516331367, −2.53341265883727031555333909839,
2.53341265883727031555333909839, 3.78664716997013961926516331367, 4.67640780536884506239351662981, 5.24479535634694968816093872884, 6.09945175507229631225492548529, 6.89452086470496642181392521129, 7.80529576046407797508604643325, 8.626116878885754625836803452995, 9.037745079856304144672572411317, 9.542573495328596472916983985115, 10.76791685486241812982375068608, 11.09594181253137038874945479137, 11.58015978794591132342486769382, 12.40808583992367980459529575016, 12.94816353122832778160077816830
