| L(s) = 1 | + 2-s − 2·7-s − 8-s − 2·9-s − 2·14-s − 16-s − 2·18-s − 2·23-s − 2·46-s + 3·49-s + 2·56-s + 4·63-s + 64-s + 2·71-s + 2·72-s + 2·79-s + 3·81-s + 3·98-s + 2·112-s + 2·113-s − 121-s + 4·126-s + 127-s + 128-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2-s − 2·7-s − 8-s − 2·9-s − 2·14-s − 16-s − 2·18-s − 2·23-s − 2·46-s + 3·49-s + 2·56-s + 4·63-s + 64-s + 2·71-s + 2·72-s + 2·79-s + 3·81-s + 3·98-s + 2·112-s + 2·113-s − 121-s + 4·126-s + 127-s + 128-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5614550726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5614550726\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.843954593408458694068303863593, −9.493651268341040971641532956191, −9.303160052920402604799326578851, −8.704452215552103367009069110378, −8.469937811031851222126383239222, −8.044919118323763129273374989878, −7.50665489231399357733332891726, −6.89467839796931541184747272690, −6.37055667514839173299920024060, −6.07680643003824312163113423068, −6.06148820894904689528403539727, −5.29672123352404281747730163147, −5.20451022881704267442759786319, −4.33499180401528034990351055655, −3.85084614058781189252424500176, −3.40098446496291043402696903894, −3.24363039444552861579197778460, −2.37007642889111239523378475339, −2.36349393864433788223460608726, −0.49418645872110009508556534167,
0.49418645872110009508556534167, 2.36349393864433788223460608726, 2.37007642889111239523378475339, 3.24363039444552861579197778460, 3.40098446496291043402696903894, 3.85084614058781189252424500176, 4.33499180401528034990351055655, 5.20451022881704267442759786319, 5.29672123352404281747730163147, 6.06148820894904689528403539727, 6.07680643003824312163113423068, 6.37055667514839173299920024060, 6.89467839796931541184747272690, 7.50665489231399357733332891726, 8.044919118323763129273374989878, 8.469937811031851222126383239222, 8.704452215552103367009069110378, 9.303160052920402604799326578851, 9.493651268341040971641532956191, 9.843954593408458694068303863593
