L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·12-s − 1.73i·13-s + (−0.5 − 0.866i)16-s − 0.999·18-s + 23-s + (−0.5 − 0.866i)24-s + 25-s + (1.49 − 0.866i)26-s − 0.999·27-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·12-s − 1.73i·13-s + (−0.5 − 0.866i)16-s − 0.999·18-s + 23-s + (−0.5 − 0.866i)24-s + 25-s + (1.49 − 0.866i)26-s − 0.999·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.212292483\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.212292483\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.73iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−11.10764694516116745440704018852, −10.37890487489081365132608418654, −9.301680230692111842299192235517, −8.548418488113028763748044728180, −7.80668251289516553993309923090, −6.80325148517004660573756113774, −5.42285308648411887814214878535, −5.00272760968178807307386165966, −3.63189507226012891018665187190, −2.91174480955689051439082182168,
1.50922728887907677219645595368, 2.59994562385008538924788678468, 3.77145034557121457510277612685, 4.87177415097243569066680883295, 6.21546761745087305879198863546, 6.91250590136447026926411748851, 8.166045166569601879076225782610, 9.215825276884303956023020781896, 9.618140212787909615951080502300, 11.20116286326183349190252404353