| L(s) = 1 | − 16·4-s + 76·13-s + 192·16-s − 320·19-s + 182·25-s − 176·43-s + 686·49-s − 1.21e3·52-s − 2.04e3·64-s − 2.19e3·67-s + 5.12e3·76-s − 2.91e3·100-s − 1.97e3·103-s − 1.69e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 62·169-s + 2.81e3·172-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.62·13-s + 3·16-s − 3.86·19-s + 1.45·25-s − 0.624·43-s + 2·49-s − 3.24·52-s − 4·64-s − 3.99·67-s + 7.72·76-s − 2.91·100-s − 1.89·103-s − 1.26·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.0282·169-s + 1.24·172-s + 0.000439·173-s + 0.000417·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8019138915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8019138915\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 182 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 1690 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 38 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 160 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 21634 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 24230 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 53170 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 1096 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 231010 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
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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
−12.94691152195925205935606848889, −12.40832657405692513383338063768, −12.04960154255104880383247793811, −10.91767910492433194611980326595, −10.53880581050305637747627321633, −10.51520954468041321658930801329, −9.579837475294514841331339077637, −8.784757697130264898296095524390, −8.711811119074366855942253021760, −8.524360527769776057367704830725, −7.75123964935835356334294521582, −6.77074334940736463532595776423, −6.14457395620300415981671198527, −5.71782231340352651384010383448, −4.77913000680829052813926151745, −4.22812039992964222842796482698, −3.99347421853276522490217423155, −2.97621542555774870073500155085, −1.59058808852255202226094972381, −0.46445386298817672160614237306,
0.46445386298817672160614237306, 1.59058808852255202226094972381, 2.97621542555774870073500155085, 3.99347421853276522490217423155, 4.22812039992964222842796482698, 4.77913000680829052813926151745, 5.71782231340352651384010383448, 6.14457395620300415981671198527, 6.77074334940736463532595776423, 7.75123964935835356334294521582, 8.524360527769776057367704830725, 8.711811119074366855942253021760, 8.784757697130264898296095524390, 9.579837475294514841331339077637, 10.51520954468041321658930801329, 10.53880581050305637747627321633, 10.91767910492433194611980326595, 12.04960154255104880383247793811, 12.40832657405692513383338063768, 12.94691152195925205935606848889
