| L(s) = 1 | + (0.809 + 0.587i)2-s + (0.688 + 2.12i)3-s + (0.309 + 0.951i)4-s + (−0.561 − 2.16i)5-s + (−0.688 + 2.12i)6-s + 0.565·7-s + (−0.309 + 0.951i)8-s + (−1.59 + 1.15i)9-s + (0.818 − 2.08i)10-s + (4.77 + 3.47i)11-s + (−1.80 + 1.31i)12-s + (4.76 − 3.46i)13-s + (0.457 + 0.332i)14-s + (4.20 − 2.68i)15-s + (−0.809 + 0.587i)16-s + (−2.16 + 6.65i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.397 + 1.22i)3-s + (0.154 + 0.475i)4-s + (−0.250 − 0.968i)5-s + (−0.281 + 0.865i)6-s + 0.213·7-s + (−0.109 + 0.336i)8-s + (−0.531 + 0.385i)9-s + (0.258 − 0.658i)10-s + (1.44 + 1.04i)11-s + (−0.520 + 0.378i)12-s + (1.32 − 0.959i)13-s + (0.122 + 0.0889i)14-s + (1.08 − 0.692i)15-s + (−0.202 + 0.146i)16-s + (−0.524 + 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0673 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0673 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82655 + 1.95404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82655 + 1.95404i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.561 + 2.16i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| good | 3 | \( 1 + (-0.688 - 2.12i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 0.565T + 7T^{2} \) |
| 11 | \( 1 + (-4.77 - 3.47i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-4.76 + 3.46i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.16 - 6.65i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (4.04 + 2.94i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.24 + 3.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.34 - 4.13i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.21 + 3.78i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.20 - 4.51i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 + (0.296 + 0.912i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.63 - 8.11i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.5 + 7.66i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.66 - 2.66i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.42 + 13.6i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.89 - 5.83i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (8.17 + 5.94i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.956 + 2.94i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.07 + 6.38i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.07 + 0.782i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.63 + 8.10i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.14595211452175196147347607176, −9.310415409729688988517820471107, −8.541884975983772186145034749057, −8.077109581958064224458669681137, −6.64586577133350639162693496660, −5.80610631588514822567563620804, −4.68220141296987959846900913602, −4.09537959810857910893702170512, −3.56178470265404589701090393270, −1.64756146224008835002465445542,
1.18097377259907931172766521602, 2.22066490697191324235551198868, 3.38257403753423553410740936425, 4.09077224562419137195617862087, 5.71669411958956574557186470175, 6.72427430765073140908439844337, 6.84534673478400242501552339750, 8.089674234022316409140158959318, 8.893078848492135731410540356795, 9.840218904082110081762372373131
