L(s) = 1 | − 6i·5-s + 6.92i·7-s + 20.7·11-s + 14i·13-s + 6·17-s + 6.92·19-s − 11·25-s + 30i·29-s − 20.7i·31-s + 41.5·35-s + 26i·37-s − 54·41-s − 20.7·43-s + 41.5i·47-s + 1.00·49-s + ⋯ |
L(s) = 1 | − 1.20i·5-s + 0.989i·7-s + 1.88·11-s + 1.07i·13-s + 0.352·17-s + 0.364·19-s − 0.440·25-s + 1.03i·29-s − 0.670i·31-s + 1.18·35-s + 0.702i·37-s − 1.31·41-s − 0.483·43-s + 0.884i·47-s + 0.0204·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.227361072\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.227361072\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 6iT - 25T^{2} \) |
| 7 | \( 1 - 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 20.7T + 121T^{2} \) |
| 13 | \( 1 - 14iT - 169T^{2} \) |
| 17 | \( 1 - 6T + 289T^{2} \) |
| 19 | \( 1 - 6.92T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 30iT - 841T^{2} \) |
| 31 | \( 1 + 20.7iT - 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 54T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 18iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 20.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 70iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 117.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 83.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 76.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 20.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 114T + 7.92e3T^{2} \) |
| 97 | \( 1 - 34T + 9.40e3T^{2} \) |
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
−9.012824043553209784234537466042, −8.512781705020459727766010490498, −7.35549819984374230546526931395, −6.48530220413066196142279280719, −5.80728262334941218987575966273, −4.86134871287296512603279605489, −4.23544502250961265704144209081, −3.21091273383519435551855237727, −1.78471396368394939442634467881, −1.15199731311521595090047502537,
0.60690585996871679307108533713, 1.75315506593520178628234586193, 3.20161174666350340422359855763, 3.59474957724721315819455328579, 4.57687490804278990761073451145, 5.78417763685452092199605828084, 6.57265078280178728174391676861, 7.09071484695040759522801777734, 7.77517741078927428306188947990, 8.728830931614808026578997963606