L(s) = 1 | + 2.41i·2-s + 2.82i·3-s − 3.82·4-s − 6.82·6-s − 2i·7-s − 4.41i·8-s − 5.00·9-s + 11-s − 10.8i·12-s + 1.17i·13-s + 4.82·14-s + 2.99·16-s + 6.82i·17-s − 12.0i·18-s + 5.65·21-s + 2.41i·22-s + ⋯ |
L(s) = 1 | + 1.70i·2-s + 1.63i·3-s − 1.91·4-s − 2.78·6-s − 0.755i·7-s − 1.56i·8-s − 1.66·9-s + 0.301·11-s − 3.12i·12-s + 0.324i·13-s + 1.29·14-s + 0.749·16-s + 1.65i·17-s − 2.84i·18-s + 1.23·21-s + 0.514i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.537451 - 0.869615i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.537451 - 0.869615i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 3 | \( 1 - 2.82iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 - 1.17iT - 13T^{2} \) |
| 17 | \( 1 - 6.82iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.17iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65iT - 97T^{2} \) |
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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
−12.85825471891489269932012584455, −11.23081722949860818297754489195, −10.34637056010627434359659982229, −9.464759372223001050603872617831, −8.663498480924810325294990729435, −7.70147510800215435011818007093, −6.46973436054866681085470473148, −5.53736346430189683680820110309, −4.42235240349404129175085364339, −3.80125312101073682934274451959,
0.824723059246608476425168087687, 2.21394959974488844257857504394, 3.02196510664762819355575242138, 4.86351319736955812035021954396, 6.25088904946254021683436791533, 7.44566097885244933517184743280, 8.609024649444682574509224626087, 9.382926957633989313241199957398, 10.62223757310466014569792610859, 11.71748719119311424796021987103