L(s) = 1 | + (0.654 + 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.118 − 0.258i)5-s + (0.142 + 0.989i)6-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (0.273 − 0.0801i)10-s + (−1.25 + 1.45i)11-s + (−0.654 + 0.755i)12-s + (−0.415 − 0.909i)14-s + (0.239 − 0.153i)15-s + (−0.959 − 0.281i)16-s + (0.239 + 1.66i)17-s + (−0.415 + 0.909i)18-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.118 − 0.258i)5-s + (0.142 + 0.989i)6-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (0.273 − 0.0801i)10-s + (−1.25 + 1.45i)11-s + (−0.654 + 0.755i)12-s + (−0.415 − 0.909i)14-s + (0.239 − 0.153i)15-s + (−0.959 − 0.281i)16-s + (0.239 + 1.66i)17-s + (−0.415 + 0.909i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.746633982\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.746633982\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-0.142 + 0.989i)T \) |
good | 5 | \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 17 | \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 29 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 31 | \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \) |
| 37 | \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \) |
| 41 | \( 1 + (-0.544 + 1.19i)T + (-0.654 - 0.755i)T^{2} \) |
| 43 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 67 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (0.698 + 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (0.654 + 0.755i)T^{2} \) |
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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
−9.613322993263367926547287621048, −8.756734064067546533038466410552, −8.076270426237968889307284926933, −7.31133352706324985843748387156, −6.65082342454040920895222379672, −5.57129565890009730345625124127, −4.66240897194516714576715946040, −4.16579833143106087325697796330, −3.02760936708889045972881287924, −2.34888416724780993664368621763,
0.938003917031618454506634926206, 2.58527670099332753674631106858, 2.95563466700887475309405099176, 3.64430023068217756215581597867, 5.03332616509712317391591116556, 5.89461202665428075180666853569, 6.51765414709454363171760141947, 7.54726352604707275754381839097, 8.382529000151050475866243210069, 9.253828670189677752955479830836