L(s) = 1 | + (0.0827 + 1.73i)3-s + (−0.532 − 3.02i)5-s + (2.03 + 1.71i)7-s + (−2.98 + 0.286i)9-s + (0.900 − 5.10i)11-s + (−0.525 − 0.191i)13-s + (5.18 − 1.17i)15-s + (−2.47 − 4.27i)17-s + (1.14 − 1.98i)19-s + (−2.79 + 3.67i)21-s + (−1.60 + 1.34i)23-s + (−4.14 + 1.50i)25-s + (−0.742 − 5.14i)27-s + (7.83 − 2.85i)29-s + (1.35 − 1.13i)31-s + ⋯ |
L(s) = 1 | + (0.0477 + 0.998i)3-s + (−0.238 − 1.35i)5-s + (0.770 + 0.646i)7-s + (−0.995 + 0.0954i)9-s + (0.271 − 1.54i)11-s + (−0.145 − 0.0530i)13-s + (1.33 − 0.302i)15-s + (−0.599 − 1.03i)17-s + (0.262 − 0.454i)19-s + (−0.609 + 0.800i)21-s + (−0.334 + 0.281i)23-s + (−0.829 + 0.301i)25-s + (−0.142 − 0.989i)27-s + (1.45 − 0.529i)29-s + (0.242 − 0.203i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32191 - 0.549099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32191 - 0.549099i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0827 - 1.73i)T \) |
good | 5 | \( 1 + (0.532 + 3.02i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.03 - 1.71i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.900 + 5.10i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.525 + 0.191i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.47 + 4.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.14 + 1.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.60 - 1.34i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.83 + 2.85i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.35 + 1.13i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.54 - 6.14i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.72 + 3.17i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.25 + 7.11i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.318 + 0.266i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 + (-1.47 - 8.37i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.90 + 6.63i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-7.91 - 2.88i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.71 - 11.6i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.03 - 1.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-14.8 + 5.39i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-14.3 + 5.23i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-4.58 + 7.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.972 - 5.51i)T + (-91.1 - 33.1i)T^{2} \) |
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
−9.919631275785432679073830236729, −8.955638768522445536768851764845, −8.644094054136824824374302176567, −7.981014870309174357931941964336, −6.34143267996095010711411327501, −5.21772876200421769269242783713, −4.92812288449790899957248264575, −3.83267158124249273492407122624, −2.59301529908939429305734427751, −0.72471477903367359221435572604,
1.56472367963419821613265523303, 2.54090366288177772486522393859, 3.82360308866385689152294503471, 4.89961475173018926518822325037, 6.46815833698227437694776507362, 6.74966432150406563437307217976, 7.69891065429706990040888343048, 8.152165068130079871454044776675, 9.508717307372396926370915662861, 10.55992532315629289348130100229