| L(s) = 1 | + (1.73 − 1.73i)3-s + (1.73 + 1.73i)7-s − 2.99i·9-s − 3.46i·11-s + (1 + i)13-s + (−1 + i)17-s + 6.92·19-s + 5.99·21-s + (−1.73 + 1.73i)23-s − 4i·29-s − 3.46i·31-s + (−5.99 − 5.99i)33-s + (5 − 5i)37-s + 3.46·39-s + 2·41-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.999i)3-s + (0.654 + 0.654i)7-s − 0.999i·9-s − 1.04i·11-s + (0.277 + 0.277i)13-s + (−0.242 + 0.242i)17-s + 1.58·19-s + 1.30·21-s + (−0.361 + 0.361i)23-s − 0.742i·29-s − 0.622i·31-s + (−1.04 − 1.04i)33-s + (0.821 − 0.821i)37-s + 0.554·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.715566950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.715566950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1.73 + 1.73i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.73 - 1.73i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + (1.73 - 1.73i)T - 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-5 + 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + (-1.73 + 1.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.73 - 1.73i)T + 47iT^{2} \) |
| 53 | \( 1 + (7 + 7i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-5.19 - 5.19i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-7 - 7i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (12.1 - 12.1i)T - 83iT^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 + (-7 + 7i)T - 97iT^{2} \) |
| 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
−9.094359819990012561493894996861, −8.280839073520432712483930274465, −7.914037691765290564224286939135, −7.08262483641580214198636251047, −6.04846957880602280422477691487, −5.36545290511056795756707207716, −3.99940537271853670163875078885, −2.99025403148042087604380891146, −2.15453422891687353979216624163, −1.09946470569192733409354580943,
1.39871670974694056980004187052, 2.75855977174473213744295854162, 3.57209127667002961174748965770, 4.56434837092910520112894175025, 4.95990579273935718112934710786, 6.35107783050154656233913908913, 7.55709922953700721295756928085, 7.83354494850774425000113750307, 8.974728612833574715336059394376, 9.434906412907538455084612562582
