| L(s) = 1 | + (−0.390 − 0.920i)2-s + (0.753 − 0.146i)3-s + (−0.694 + 0.719i)4-s + (2.20 − 0.348i)5-s + (−0.429 − 0.636i)6-s + (−1.29 + 4.83i)7-s + (0.933 + 0.358i)8-s + (−2.23 + 0.902i)9-s + (−1.18 − 1.89i)10-s + (1.30 + 0.277i)11-s + (−0.418 + 0.643i)12-s + (−1.27 + 0.156i)13-s + (4.95 − 0.696i)14-s + (1.61 − 0.586i)15-s + (−0.0348 − 0.999i)16-s + (1.86 − 3.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.276 − 0.650i)2-s + (0.435 − 0.0845i)3-s + (−0.347 + 0.359i)4-s + (0.987 − 0.155i)5-s + (−0.175 − 0.259i)6-s + (−0.489 + 1.82i)7-s + (0.330 + 0.126i)8-s + (−0.744 + 0.300i)9-s + (−0.374 − 0.599i)10-s + (0.394 + 0.0838i)11-s + (−0.120 + 0.185i)12-s + (−0.353 + 0.0433i)13-s + (1.32 − 0.186i)14-s + (0.416 − 0.151i)15-s + (−0.00872 − 0.249i)16-s + (0.452 − 0.752i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17983 + 0.633030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17983 + 0.633030i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.390 + 0.920i)T \) |
| 5 | \( 1 + (-2.20 + 0.348i)T \) |
| 19 | \( 1 + (4.34 - 0.293i)T \) |
| good | 3 | \( 1 + (-0.753 + 0.146i)T + (2.78 - 1.12i)T^{2} \) |
| 7 | \( 1 + (1.29 - 4.83i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.30 - 0.277i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.27 - 0.156i)T + (12.6 - 3.14i)T^{2} \) |
| 17 | \( 1 + (-1.86 + 3.10i)T + (-7.98 - 15.0i)T^{2} \) |
| 23 | \( 1 + (2.61 - 8.56i)T + (-19.0 - 12.8i)T^{2} \) |
| 29 | \( 1 + (1.72 - 6.90i)T + (-25.6 - 13.6i)T^{2} \) |
| 31 | \( 1 + (1.24 + 0.130i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.97 - 5.83i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-7.53 + 0.263i)T + (40.9 - 2.86i)T^{2} \) |
| 43 | \( 1 + (2.58 + 3.68i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (2.74 - 1.65i)T + (22.0 - 41.4i)T^{2} \) |
| 53 | \( 1 + (0.0701 + 4.02i)T + (-52.9 + 1.84i)T^{2} \) |
| 59 | \( 1 + (-5.84 + 3.65i)T + (25.8 - 53.0i)T^{2} \) |
| 61 | \( 1 + (-3.52 + 1.87i)T + (34.1 - 50.5i)T^{2} \) |
| 67 | \( 1 + (-2.42 + 2.78i)T + (-9.32 - 66.3i)T^{2} \) |
| 71 | \( 1 + (1.33 - 0.652i)T + (43.7 - 55.9i)T^{2} \) |
| 73 | \( 1 + (-16.1 - 1.98i)T + (70.8 + 17.6i)T^{2} \) |
| 79 | \( 1 + (5.12 - 7.60i)T + (-29.5 - 73.2i)T^{2} \) |
| 83 | \( 1 + (-3.84 - 4.74i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 + (0.558 - 15.9i)T + (-88.7 - 6.20i)T^{2} \) |
| 97 | \( 1 + (-10.6 + 9.27i)T + (13.4 - 96.0i)T^{2} \) |
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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
−9.744988150135972355995520661165, −9.452205624011406960851159567568, −8.771989928761433932080006184609, −8.051322613325317160544194882407, −6.68410118886073415203745933805, −5.63768337862145489978956667275, −5.17207040714107749973238528570, −3.40118054152623349753030032308, −2.53342019696399137838092211812, −1.83774372608931175911061318452,
0.62309596011876601736663792884, 2.31130408069831505587384776026, 3.71529801998271217516573018244, 4.50166923213422051209688174611, 6.05687476773865285538041516545, 6.35519834192399268119702327817, 7.37282273249388178524705918986, 8.218015971009119844253602168860, 9.079647764197819965887660975169, 9.926179087690089161192801566476
