| L(s) = 1 | − 0.356i·3-s + (0.376 + 2.20i)5-s − 2.46i·7-s + 2.87·9-s − 1.61·11-s + 2.62i·13-s + (0.785 − 0.134i)15-s − 2.58i·17-s + 4.02·19-s − 0.878·21-s + i·23-s + (−4.71 + 1.65i)25-s − 2.09i·27-s + 7.08·29-s + 4.58·31-s + ⋯ |
| L(s) = 1 | − 0.205i·3-s + (0.168 + 0.985i)5-s − 0.931i·7-s + 0.957·9-s − 0.487·11-s + 0.728i·13-s + (0.202 − 0.0346i)15-s − 0.627i·17-s + 0.923·19-s − 0.191·21-s + 0.208i·23-s + (−0.943 + 0.331i)25-s − 0.402i·27-s + 1.31·29-s + 0.823·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.945617418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.945617418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.376 - 2.20i)T \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 0.356iT - 3T^{2} \) |
| 7 | \( 1 + 2.46iT - 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 + 2.58iT - 17T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 - 2.96iT - 37T^{2} \) |
| 41 | \( 1 + 5.71T + 41T^{2} \) |
| 43 | \( 1 + 2.30iT - 43T^{2} \) |
| 47 | \( 1 + 6.88iT - 47T^{2} \) |
| 53 | \( 1 - 6.76iT - 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 - 15.7iT - 67T^{2} \) |
| 71 | \( 1 - 5.25T + 71T^{2} \) |
| 73 | \( 1 + 6.03iT - 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 + 8.59iT - 83T^{2} \) |
| 89 | \( 1 - 8.84T + 89T^{2} \) |
| 97 | \( 1 - 8.01iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.516603432748187574469965073750, −8.342436354518774161322259008485, −7.38241445499749635108663865194, −7.05044344605818692344243099668, −6.36447390762928507531299313130, −5.14764936510613672637445536639, −4.25906020581943831644939170828, −3.34700072338901225887664772828, −2.31372077556602765889268833238, −1.03639338196945115734507379872,
0.957199736531995539253762338435, 2.15837395442535619269949189105, 3.30850843939809449154352407676, 4.47854347349832476659860587417, 5.14183522991173194926363540234, 5.83036083446075592013654052219, 6.80400311800911836321388509134, 7.978751080362969553855633095730, 8.348813059503281253471584347745, 9.289286534754416873716341561222
