L(s) = 1 | + (−0.707 − 0.707i)2-s + (−2.87 − 0.854i)3-s − 3i·4-s + (−4.57 + 2.01i)5-s + (1.42 + 2.63i)6-s + (3.94 + 5.77i)7-s + (−4.94 + 4.94i)8-s + (7.53 + 4.91i)9-s + (4.65 + 1.81i)10-s + 2.58i·11-s + (−2.56 + 8.62i)12-s + (−8.94 + 8.94i)13-s + (1.29 − 6.87i)14-s + (14.8 − 1.86i)15-s − 4.99·16-s + (0.581 − 0.581i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.353i)2-s + (−0.958 − 0.284i)3-s − 0.750i·4-s + (−0.915 + 0.402i)5-s + (0.238 + 0.439i)6-s + (0.564 + 0.825i)7-s + (−0.618 + 0.618i)8-s + (0.837 + 0.546i)9-s + (0.465 + 0.181i)10-s + 0.235i·11-s + (−0.213 + 0.718i)12-s + (−0.687 + 0.687i)13-s + (0.0924 − 0.491i)14-s + (0.992 − 0.124i)15-s − 0.312·16-s + (0.0342 − 0.0342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.171684 + 0.203816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.171684 + 0.203816i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.87 + 0.854i)T \) |
| 5 | \( 1 + (4.57 - 2.01i)T \) |
| 7 | \( 1 + (-3.94 - 5.77i)T \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + 4iT^{2} \) |
| 11 | \( 1 - 2.58iT - 121T^{2} \) |
| 13 | \( 1 + (8.94 - 8.94i)T - 169iT^{2} \) |
| 17 | \( 1 + (-0.581 + 0.581i)T - 289iT^{2} \) |
| 19 | \( 1 + 16.5T + 361T^{2} \) |
| 23 | \( 1 + (26.5 - 26.5i)T - 529iT^{2} \) |
| 29 | \( 1 - 11.5T + 841T^{2} \) |
| 31 | \( 1 + 30.8iT - 961T^{2} \) |
| 37 | \( 1 + (41.4 - 41.4i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 17.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (25.1 + 25.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-45.2 + 45.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (34.1 - 34.1i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 47.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 78.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-72.4 + 72.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 49.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (26.0 - 26.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 75.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (53.6 + 53.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 14.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-25.9 - 25.9i)T + 9.40e3iT^{2} \) |
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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
−13.86494274221510656664788780357, −12.07104749541768982510179398844, −11.80632974266075515300970747944, −10.80917448915012641772671297482, −9.810563423344152454230035010269, −8.367984216303151686652981069349, −7.02235572470009961547827578622, −5.78645893628489852851134024101, −4.56333742963854897160008816129, −1.98902820466104370209013269539,
0.24373590310307137208782387830, 3.80567008954487713144581923863, 4.84177291241758311034565264287, 6.65189029205087850561324072658, 7.69060432712437362407731593527, 8.581140693879308854001308571849, 10.18796318697728442935747647955, 11.15751579506757961381292746085, 12.24792539681856457294983459923, 12.76527564373619118974044656912