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
−9.458662684714851864404080516706, −8.621809818729676507247985036694, −7.920019884955751442423591008300, −7.13867389319388351197198050246, −6.00586804970885833955879536878, −5.46693619519736486074087230697, −4.45586560799290528528078787307, −3.39485230222342112397070753471, −2.47721962337422856461379687633, −0.985410814038974510103190904706,
0.77448604197313909151175145249, 2.35000714234537269357006211758, 3.39884688204924552232079268052, 4.22556459238636769494077186938, 5.18225187955212691209606686313, 6.32755156734376256877235140711, 6.89646202038817991171433881562, 7.71566523213339628157756761594, 8.490249047453280027012231826294, 9.444033007247618621045438384566