L(s) = 1 | + (−0.5 + 0.866i)5-s + (−1 − 1.73i)7-s + (−1 + 1.73i)13-s + 6·17-s − 4·19-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + (3 + 5.19i)29-s + (2 − 3.46i)31-s + 1.99·35-s + 2·37-s + (3 − 5.19i)41-s + (5 + 8.66i)43-s + (−3 − 5.19i)47-s + (1.50 − 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.277 + 0.480i)13-s + 1.45·17-s − 0.917·19-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + (0.557 + 0.964i)29-s + (0.359 − 0.622i)31-s + 0.338·35-s + 0.328·37-s + (0.468 − 0.811i)41-s + (0.762 + 1.32i)43-s + (−0.437 − 0.757i)47-s + (0.214 − 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.516110132\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.516110132\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.444033007247618621045438384566, −8.490249047453280027012231826294, −7.71566523213339628157756761594, −6.89646202038817991171433881562, −6.32755156734376256877235140711, −5.18225187955212691209606686313, −4.22556459238636769494077186938, −3.39884688204924552232079268052, −2.35000714234537269357006211758, −0.77448604197313909151175145249,
0.985410814038974510103190904706, 2.47721962337422856461379687633, 3.39485230222342112397070753471, 4.45586560799290528528078787307, 5.46693619519736486074087230697, 6.00586804970885833955879536878, 7.13867389319388351197198050246, 7.920019884955751442423591008300, 8.621809818729676507247985036694, 9.458662684714851864404080516706