| L(s) = 1 | + (0.915 + 1.07i)2-s + (−0.290 + 0.956i)3-s + (−0.322 + 1.97i)4-s + (−0.270 + 2.74i)5-s + (−1.29 + 0.563i)6-s + (−1.85 − 2.77i)7-s + (−2.42 + 1.46i)8-s + (−0.831 − 0.555i)9-s + (−3.20 + 2.22i)10-s + (4.99 + 2.66i)11-s + (−1.79 − 0.881i)12-s + (−3.79 + 0.374i)13-s + (1.29 − 4.53i)14-s + (−2.54 − 1.05i)15-s + (−3.79 − 1.27i)16-s + (−0.856 + 0.354i)17-s + ⋯ |
| L(s) = 1 | + (0.647 + 0.761i)2-s + (−0.167 + 0.552i)3-s + (−0.161 + 0.986i)4-s + (−0.120 + 1.22i)5-s + (−0.529 + 0.230i)6-s + (−0.700 − 1.04i)7-s + (−0.856 + 0.516i)8-s + (−0.277 − 0.185i)9-s + (−1.01 + 0.702i)10-s + (1.50 + 0.804i)11-s + (−0.518 − 0.254i)12-s + (−1.05 + 0.103i)13-s + (0.345 − 1.21i)14-s + (−0.657 − 0.272i)15-s + (−0.948 − 0.317i)16-s + (−0.207 + 0.0860i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.162146 + 1.45160i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.162146 + 1.45160i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.915 - 1.07i)T \) |
| 3 | \( 1 + (0.290 - 0.956i)T \) |
| good | 5 | \( 1 + (0.270 - 2.74i)T + (-4.90 - 0.975i)T^{2} \) |
| 7 | \( 1 + (1.85 + 2.77i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.99 - 2.66i)T + (6.11 + 9.14i)T^{2} \) |
| 13 | \( 1 + (3.79 - 0.374i)T + (12.7 - 2.53i)T^{2} \) |
| 17 | \( 1 + (0.856 - 0.354i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.48 - 2.03i)T + (3.70 + 18.6i)T^{2} \) |
| 23 | \( 1 + (1.11 + 5.59i)T + (-21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-2.13 - 3.99i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (-6.37 - 6.37i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.01 + 6.11i)T + (-7.21 + 36.2i)T^{2} \) |
| 41 | \( 1 + (0.359 - 0.0715i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.25 - 10.7i)T + (-35.7 + 23.8i)T^{2} \) |
| 47 | \( 1 + (-3.33 - 8.04i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.428 - 0.802i)T + (-29.4 - 44.0i)T^{2} \) |
| 59 | \( 1 + (-0.295 - 0.0291i)T + (57.8 + 11.5i)T^{2} \) |
| 61 | \( 1 + (-10.5 - 3.19i)T + (50.7 + 33.8i)T^{2} \) |
| 67 | \( 1 + (7.60 + 2.30i)T + (55.7 + 37.2i)T^{2} \) |
| 71 | \( 1 + (-9.07 + 6.06i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.960 - 1.43i)T + (-27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-3.98 + 9.62i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-9.68 + 11.8i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (0.373 - 1.87i)T + (-82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-5.88 - 5.88i)T + 97iT^{2} \) |
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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
−11.96044227461390303898437033272, −10.77348565808689568364233491861, −10.00112324600442854408439282110, −9.092660234447984109781090933188, −7.58498707659901367272604724737, −6.79367103775611837113500624393, −6.40755694167855506090698331406, −4.73002593040089147369658578968, −3.91262064377299349460322485474, −2.93556765789158534541637526246,
0.834896008257784391687746633526, 2.37450833715834972879175534176, 3.75764132165778299803520356549, 5.06052256197369511611919327968, 5.82767020198965257977144142471, 6.80200783302131051393789034844, 8.459865251924565625578944630774, 9.220660818997293091339694201451, 9.849950322999013581071451053646, 11.54368933332736140165169666805
