| L(s) = 1 | − i·2-s − 4-s − 5-s + 2.68i·7-s + i·8-s + i·10-s − 1.73·11-s − 2.55·13-s + 2.68·14-s + 16-s − 1.27·17-s + 1.40i·19-s + 20-s + 1.73i·22-s + (4.59 − 1.35i)23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447·5-s + 1.01i·7-s + 0.353i·8-s + 0.316i·10-s − 0.521·11-s − 0.708·13-s + 0.718·14-s + 0.250·16-s − 0.308·17-s + 0.322i·19-s + 0.223·20-s + 0.369i·22-s + (0.958 − 0.283i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8243884591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8243884591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-4.59 + 1.35i)T \) |
| good | 7 | \( 1 - 2.68iT - 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 19 | \( 1 - 1.40iT - 19T^{2} \) |
| 29 | \( 1 + 9.62iT - 29T^{2} \) |
| 31 | \( 1 + 1.47T + 31T^{2} \) |
| 37 | \( 1 + 4.20iT - 37T^{2} \) |
| 41 | \( 1 + 3.40iT - 41T^{2} \) |
| 43 | \( 1 - 2.35iT - 43T^{2} \) |
| 47 | \( 1 + 3.21iT - 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 + 7.74iT - 59T^{2} \) |
| 61 | \( 1 + 6.78iT - 61T^{2} \) |
| 67 | \( 1 + 7.71iT - 67T^{2} \) |
| 71 | \( 1 + 4.44iT - 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 10.0iT - 79T^{2} \) |
| 83 | \( 1 - 8.67T + 83T^{2} \) |
| 89 | \( 1 - 3.72T + 89T^{2} \) |
| 97 | \( 1 + 7.03iT - 97T^{2} \) |
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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
−8.970460618975119731050282582210, −8.171800645661346691863371833430, −7.49917659552998032091651841237, −6.40250277794341887783079884433, −5.44587165138197125862276279951, −4.77390078899928159523150145495, −3.78063091105328628228840353722, −2.72993466009666545716081346572, −2.06521002000647397402969216571, −0.33500430293841913534665684759,
1.08865995719968532047082567625, 2.80407973642384884818908420701, 3.78747899586088298592979379198, 4.71747431346302441527536917356, 5.26671455295819119314011740618, 6.47569229936701237070878237267, 7.28671094252363425621484348700, 7.49897909369108070677518572086, 8.574587864806215716728625658367, 9.182864228226533680850596854746
