| L(s) = 1 | + (0.142 + 0.989i)3-s + (−2.42 − 0.713i)5-s + (−2.20 + 2.54i)7-s + (−0.959 + 0.281i)9-s + (−3.45 + 2.22i)11-s + (0.0587 + 0.0678i)13-s + (0.360 − 2.50i)15-s + (−0.336 + 0.736i)17-s + (2.46 + 5.40i)19-s + (−2.83 − 1.81i)21-s + (−0.531 − 4.76i)23-s + (1.18 + 0.760i)25-s + (−0.415 − 0.909i)27-s + (−0.110 + 0.242i)29-s + (−0.0278 + 0.193i)31-s + ⋯ |
| L(s) = 1 | + (0.0821 + 0.571i)3-s + (−1.08 − 0.318i)5-s + (−0.832 + 0.961i)7-s + (−0.319 + 0.0939i)9-s + (−1.04 + 0.670i)11-s + (0.0162 + 0.0188i)13-s + (0.0930 − 0.646i)15-s + (−0.0815 + 0.178i)17-s + (0.566 + 1.23i)19-s + (−0.617 − 0.396i)21-s + (−0.110 − 0.993i)23-s + (0.236 + 0.152i)25-s + (−0.0799 − 0.175i)27-s + (−0.0205 + 0.0450i)29-s + (−0.00500 + 0.0348i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.152761 + 0.528813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.152761 + 0.528813i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (0.531 + 4.76i)T \) |
| good | 5 | \( 1 + (2.42 + 0.713i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (2.20 - 2.54i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (3.45 - 2.22i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.0587 - 0.0678i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.336 - 0.736i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.46 - 5.40i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.110 - 0.242i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.0278 - 0.193i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.79 + 0.527i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (3.91 + 1.14i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.71 - 11.9i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 9.27T + 47T^{2} \) |
| 53 | \( 1 + (5.97 - 6.89i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (2.56 + 2.96i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.79 + 12.5i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-12.6 - 8.15i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-8.24 - 5.29i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.58 - 3.46i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (11.1 + 12.8i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (5.49 - 1.61i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.638 + 4.44i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.43 - 0.715i)T + (81.6 + 52.4i)T^{2} \) |
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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
−12.42900830991215075741144529888, −11.39254763400017242095564763919, −10.27832919768261581513626031778, −9.484536367063353888151458501937, −8.396779460996532004029562288059, −7.66865294795323969657495543995, −6.18776946639097090454084367892, −5.05299021161832411958008522641, −3.91600940161085679068370367245, −2.68156195724179456374186427722,
0.39919277799978554629412178504, 2.91822027452646321291907436303, 3.87130570130556311751261369867, 5.45157128322755400681633864575, 6.91441225225568588771816898616, 7.41953467623217067433529459377, 8.366560590530700665717566515588, 9.640775744973002959520359196639, 10.77802505205139613043659615614, 11.42241197501430956902062606453
