L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (−1.70 + 2.95i)11-s + (0.499 + 0.866i)12-s + 0.999·15-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s + (−0.499 + 0.866i)18-s + (1.41 + 2.44i)19-s − 0.999·20-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.514 + 0.891i)11-s + (0.144 + 0.249i)12-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.171 + 0.297i)17-s + (−0.117 + 0.204i)18-s + (0.324 + 0.561i)19-s − 0.223·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.183668133\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183668133\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (0.707 - 1.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.414 - 0.717i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 + (4.53 - 7.85i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.707 + 1.22i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 + (-2.53 - 4.39i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.65 - 11.5i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.24 - 12.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.171 + 0.297i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.94 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + (1.82 - 3.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.65 + 9.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + (-5.24 - 9.08i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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
−9.463545801076700489384994801030, −9.014196874026487006450353350191, −7.83636492918261762718205637999, −7.45342189021668347562444862163, −6.47493504717400522196668687869, −5.45568883557095004276687322825, −4.33942250385754095274347934415, −3.25413934576986458118547600675, −2.35356893214594785948174154581, −1.39027138690386347039920585880,
0.52611030048546418460169902068, 2.22719185993947789928075511618, 3.43270072598801718893289850816, 4.54036799671397139120768088094, 5.36919440661996387875593296889, 6.04838804013094700368736240085, 7.12810469369161009240860317043, 7.980680836433718063210783897643, 8.621086309442217681683081684942, 9.377741144118413939796833008656