L(s) = 1 | + (−2.12 + 0.707i)5-s + (3.46 − 3.46i)7-s − 4.89i·11-s + (−1 − i)13-s + (2.82 + 2.82i)17-s − 6.92i·19-s + (3.99 − 3i)25-s − 7.07·29-s − 6.92·31-s + (−4.89 + 9.79i)35-s + (−1 + i)37-s + 9.89i·41-s + (6.92 + 6.92i)43-s + (−4.89 − 4.89i)47-s − 16.9i·49-s + ⋯ |
L(s) = 1 | + (−0.948 + 0.316i)5-s + (1.30 − 1.30i)7-s − 1.47i·11-s + (−0.277 − 0.277i)13-s + (0.685 + 0.685i)17-s − 1.58i·19-s + (0.799 − 0.600i)25-s − 1.31·29-s − 1.24·31-s + (−0.828 + 1.65i)35-s + (−0.164 + 0.164i)37-s + 1.54i·41-s + (1.05 + 1.05i)43-s + (−0.714 − 0.714i)47-s − 2.42i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.167674827\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167674827\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.12 - 0.707i)T \) |
good | 7 | \( 1 + (-3.46 + 3.46i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.89iT - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (-6.92 - 6.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.89 + 4.89i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.82 - 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (6.92 - 6.92i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (7 + 7i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + (-4.89 + 4.89i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 5i)T - 97iT^{2} \) |
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
−8.324348432004921089221770478617, −7.63115947894520513261580584164, −7.32144165950187410383649881441, −6.25884853049551782065386983837, −5.24835486822132445063127042275, −4.47007728058936098700534796457, −3.71835029251619202970537247448, −2.96894853134270743437548813413, −1.40589291838087897196198671755, −0.38479547858160083146220780828,
1.59683943184660485812465870582, 2.24464185671693851951835457702, 3.63574826164251829512715067312, 4.41445634073735049190764080033, 5.26882474198170825443262653968, 5.64891286877704524203057430781, 7.16419239392334339925400388377, 7.58601200003251497765531431629, 8.210480338012072510927162665280, 9.054499478756534698224274099441