L(s) = 1 | − 3-s + 9-s + 2·11-s − 25-s − 27-s − 2·33-s − 49-s − 2·59-s − 2·73-s + 75-s + 81-s + 2·83-s + 2·97-s + 2·99-s − 2·107-s + ⋯ |
L(s) = 1 | − 3-s + 9-s + 2·11-s − 25-s − 27-s − 2·33-s − 49-s − 2·59-s − 2·73-s + 75-s + 81-s + 2·83-s + 2·97-s + 2·99-s − 2·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6758275560\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6758275560\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.70957848674615572709908029269, −10.80323671458423038130760524037, −9.755549068104677975073807155194, −9.033119266802489769096367800822, −7.64700335958132785483447087494, −6.60071251389104815841797533111, −5.97981258635214669939844358455, −4.66487465990651944039652242126, −3.71113239212401521902242330066, −1.53070354608733596024417520270,
1.53070354608733596024417520270, 3.71113239212401521902242330066, 4.66487465990651944039652242126, 5.97981258635214669939844358455, 6.60071251389104815841797533111, 7.64700335958132785483447087494, 9.033119266802489769096367800822, 9.755549068104677975073807155194, 10.80323671458423038130760524037, 11.70957848674615572709908029269