L(s) = 1 | + (2.09 − 1.20i)2-s + (−0.5 − 0.866i)3-s + (1.91 − 3.31i)4-s − 2.82i·5-s + (−2.09 − 1.20i)6-s + (2.44 + 1.41i)7-s − 4.41i·8-s + (−0.499 + 0.866i)9-s + (−3.41 − 5.91i)10-s + (−1.73 + i)11-s − 3.82·12-s + 6.82·14-s + (−2.44 + 1.41i)15-s + (−1.49 − 2.59i)16-s + (−1.82 + 3.16i)17-s + 2.41i·18-s + ⋯ |
L(s) = 1 | + (1.47 − 0.853i)2-s + (−0.288 − 0.499i)3-s + (0.957 − 1.65i)4-s − 1.26i·5-s + (−0.853 − 0.492i)6-s + (0.925 + 0.534i)7-s − 1.56i·8-s + (−0.166 + 0.288i)9-s + (−1.07 − 1.87i)10-s + (−0.522 + 0.301i)11-s − 1.10·12-s + 1.82·14-s + (−0.632 + 0.365i)15-s + (−0.374 − 0.649i)16-s + (−0.443 + 0.768i)17-s + 0.569i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40409 - 2.66449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40409 - 2.66449i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-2.09 + 1.20i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + (-2.44 - 1.41i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.73 - i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.82 - 3.16i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.44 - 1.41i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.82iT - 31T^{2} \) |
| 37 | \( 1 + (-3.16 + 1.82i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (9.37 - 5.41i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.82 + 8.36i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 0.343iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (-3.16 - 1.82i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.65 + 8.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.01 - 0.585i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.73 - i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 7.65iT - 83T^{2} \) |
| 89 | \( 1 + (-7.94 + 4.58i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.63 + 3.82i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.02547441584754033677015797444, −10.11245362123454406064172204815, −8.710121613793257120954199589106, −8.002457160837505231225127715313, −6.48690854330236027643334806268, −5.32449938750639606516318567972, −5.03774233795189076786258082833, −3.97407843328153901602278722219, −2.34534312871710837644555569749, −1.40671411468799221530122404930,
2.70342942505642475939738294589, 3.69664969727484417800221013028, 4.69664819928250812370452923049, 5.50499349158747659982729283386, 6.48222526633183684963916065709, 7.32544768982263945892440317558, 7.959307495474695256602705560637, 9.580703820489487358772297609580, 10.72867977191733125535980074964, 11.30965470015594975845275215483