L(s) = 1 | + (−1.37 − 2.37i)5-s + (−0.362 − 0.209i)11-s + (1.32 − 0.765i)13-s − 3.90·17-s − 5.91i·19-s + (7.72 − 4.46i)23-s + (−1.26 + 2.18i)25-s + (−6.00 − 3.46i)29-s + (−3.05 + 1.76i)31-s + 9.09·37-s + (−1.06 − 1.84i)41-s + (−5.77 + 10.0i)43-s + (−0.885 + 1.53i)47-s − 3.92i·53-s + 1.14i·55-s + ⋯ |
L(s) = 1 | + (−0.613 − 1.06i)5-s + (−0.109 − 0.0630i)11-s + (0.367 − 0.212i)13-s − 0.947·17-s − 1.35i·19-s + (1.61 − 0.930i)23-s + (−0.252 + 0.437i)25-s + (−1.11 − 0.643i)29-s + (−0.548 + 0.316i)31-s + 1.49·37-s + (−0.165 − 0.287i)41-s + (−0.881 + 1.52i)43-s + (−0.129 + 0.223i)47-s − 0.538i·53-s + 0.154i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7376040499\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7376040499\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.37 + 2.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.362 + 0.209i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 0.765i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + 5.91iT - 19T^{2} \) |
| 23 | \( 1 + (-7.72 + 4.46i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.00 + 3.46i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.05 - 1.76i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.09T + 37T^{2} \) |
| 41 | \( 1 + (1.06 + 1.84i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.77 - 10.0i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.885 - 1.53i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3.92iT - 53T^{2} \) |
| 59 | \( 1 + (2.02 + 3.51i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.61 + 0.932i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.38 - 11.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.51iT - 71T^{2} \) |
| 73 | \( 1 + 1.90iT - 73T^{2} \) |
| 79 | \( 1 + (-0.433 + 0.751i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.45 - 5.99i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 + (-0.200 - 0.115i)T + (48.5 + 84.0i)T^{2} \) |
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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
−7.988706145119895964031407322003, −7.09206980043416716315477376857, −6.49373976296210678140875211582, −5.50584191508021778992084409518, −4.69592695843574937546072313387, −4.38754651921510022171781419877, −3.27647535614877871048755497497, −2.39228733157504724509585600785, −1.11243070327360138959586710890, −0.21596895766333132830586939265,
1.44248458695621271207303501535, 2.48046948421227823451639346234, 3.46433307759090803252870978516, 3.84052670718823175082784707854, 4.92841335441457425711767613287, 5.75899642433262742868982482238, 6.52894303759192044030014732076, 7.24656119203118555563987824587, 7.61128556182003323264181871684, 8.550935638618960388582045690306