| L(s) = 1 | + (−0.707 + 0.707i)3-s + (1.87 − 1.87i)7-s − 1.00i·9-s − 2·11-s + (−1.41 + 1.41i)13-s + (−2.82 − 2.82i)17-s − 5.29·19-s + 2.64i·21-s + (3.74 + 3.74i)23-s + (0.707 + 0.707i)27-s + 6i·29-s + 5.29i·31-s + (1.41 − 1.41i)33-s − 2.00i·39-s + 10.5i·41-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.707 − 0.707i)7-s − 0.333i·9-s − 0.603·11-s + (−0.392 + 0.392i)13-s + (−0.685 − 0.685i)17-s − 1.21·19-s + 0.577i·21-s + (0.780 + 0.780i)23-s + (0.136 + 0.136i)27-s + 1.11i·29-s + 0.950i·31-s + (0.246 − 0.246i)33-s − 0.320i·39-s + 1.65i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7898124089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7898124089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.87 + 1.87i)T \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (1.41 - 1.41i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.82 + 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + (-3.74 - 3.74i)T + 23iT^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 5.29iT - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 10.5iT - 41T^{2} \) |
| 43 | \( 1 + (3.74 + 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.74 - 3.74i)T + 53iT^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + (-3.74 + 3.74i)T - 67iT^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (-9.89 + 9.89i)T - 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-1.41 - 1.41i)T + 97iT^{2} \) |
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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
−9.306320546203464060822317554436, −8.714288730783085475034996569220, −7.71681374534171147623769143204, −7.05510823376052148868469987673, −6.28201227845912770793420090194, −5.00930325540261471750981954403, −4.79299383337817143455838614669, −3.74713196489849050656120661986, −2.59207581103366388181601252934, −1.29544757703803999667846046628,
0.29668804491520888916228260818, 1.97592587516224622648029948962, 2.55818485413719840984859380903, 4.09206023217097764269607957708, 4.91105186164542602644693842112, 5.69240229842485286734104528510, 6.40773621403425552081619457793, 7.29480356373383504100640652894, 8.220657969061639797913611067610, 8.553740048007937757342871687594
