L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1 + 1.41i)3-s + 0.999i·4-s + (1.41 − 1.41i)5-s + (−0.292 − 1.70i)6-s + (1 − i)7-s + (−2.12 − 2.12i)8-s + (−1.00 − 2.82i)9-s + 2.00i·10-s + (2.82 + 2.82i)11-s + (−1.41 − 0.999i)12-s + (−2 − 3i)13-s + 1.41i·14-s + (0.585 + 3.41i)15-s + 1.00·16-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.577 + 0.816i)3-s + 0.499i·4-s + (0.632 − 0.632i)5-s + (−0.119 − 0.696i)6-s + (0.377 − 0.377i)7-s + (−0.750 − 0.750i)8-s + (−0.333 − 0.942i)9-s + 0.632i·10-s + (0.852 + 0.852i)11-s + (−0.408 − 0.288i)12-s + (−0.554 − 0.832i)13-s + 0.377i·14-s + (0.151 + 0.881i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.468167 + 0.337532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.468167 + 0.337532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1 - 1.41i)T \) |
| 13 | \( 1 + (2 + 3i)T \) |
good | 2 | \( 1 + (0.707 - 0.707i)T - 2iT^{2} \) |
| 5 | \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (-1 - i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + (5 + 5i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.41 + 1.41i)T - 41iT^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + (-2.82 - 2.82i)T + 47iT^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (-2.82 - 2.82i)T + 59iT^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (5 + 5i)T + 67iT^{2} \) |
| 71 | \( 1 + (2.82 - 2.82i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + (-5.65 + 5.65i)T - 83iT^{2} \) |
| 89 | \( 1 + (9.89 + 9.89i)T + 89iT^{2} \) |
| 97 | \( 1 + (-7 - 7i)T + 97iT^{2} \) |
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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
−16.78143769831990227594536998304, −15.69877238775338654574605751273, −14.51485899582245118845021311444, −12.78060801443246797948558853327, −11.77109277835793986615228769704, −10.02155482040305598488882210727, −9.211180387486936601493784758346, −7.64074234927874492951923188733, −5.91663455141999524000695565236, −4.20924296412302041216682511933,
2.01645263655581689352851342192, 5.60369323473452934674015467315, 6.70588095140817211226667910466, 8.664988877066061889315090811529, 10.08889150269282229426962884314, 11.27655515850053202120672354428, 12.04534595841437143990557330796, 13.93288389298878883984517220529, 14.46714382280142753892680151742, 16.38478840930367891237437276030