| L(s) = 1 | + (−1.30 − 0.549i)2-s + (−0.988 + 0.149i)3-s + (1.39 + 1.43i)4-s + (−2.67 − 1.05i)5-s + (1.37 + 0.348i)6-s + (−0.927 + 2.47i)7-s + (−1.03 − 2.63i)8-s + (0.955 − 0.294i)9-s + (2.90 + 2.83i)10-s + (−1.21 + 3.93i)11-s + (−1.59 − 1.20i)12-s + (4.15 − 0.948i)13-s + (2.57 − 2.71i)14-s + (2.80 + 0.639i)15-s + (−0.0991 + 3.99i)16-s + (−1.41 − 2.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.921 − 0.388i)2-s + (−0.570 + 0.0860i)3-s + (0.698 + 0.715i)4-s + (−1.19 − 0.469i)5-s + (0.559 + 0.142i)6-s + (−0.350 + 0.936i)7-s + (−0.365 − 0.930i)8-s + (0.318 − 0.0982i)9-s + (0.920 + 0.897i)10-s + (−0.365 + 1.18i)11-s + (−0.460 − 0.348i)12-s + (1.15 − 0.262i)13-s + (0.686 − 0.726i)14-s + (0.723 + 0.165i)15-s + (−0.0247 + 0.999i)16-s + (−0.342 − 0.502i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.241027 - 0.273298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.241027 - 0.273298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 + 0.549i)T \) |
| 3 | \( 1 + (0.988 - 0.149i)T \) |
| 7 | \( 1 + (0.927 - 2.47i)T \) |
| good | 5 | \( 1 + (2.67 + 1.05i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (1.21 - 3.93i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-4.15 + 0.948i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (1.41 + 2.07i)T + (-6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (1.77 + 3.07i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0806 - 0.118i)T + (-8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (1.88 + 0.905i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.09 + 3.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.213 + 2.84i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-5.74 + 4.58i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-0.530 - 0.422i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-4.03 + 3.74i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.987 + 13.1i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (0.259 + 0.661i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (8.24 - 0.617i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (13.2 + 7.66i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.00 - 12.4i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-8.63 + 9.30i)T + (-5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (-0.803 + 0.463i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.42 + 15.0i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-4.14 - 13.4i)T + (-73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 - 1.93iT - 97T^{2} \) |
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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
−10.55989931959110970942866460478, −9.481987805937519517897178661724, −8.807016443026779739591183887681, −7.934716198921215129390536365510, −7.10044872518084991265908427539, −6.05582749977515526943765820777, −4.71058691516972008006198003228, −3.65589336390072172935826807409, −2.22519980550406112752422249530, −0.36307312004229800848581699242,
1.07709204594009389902807317733, 3.23478167397751177179063912692, 4.27010853010265346324967524593, 5.92424418017477004495064803845, 6.49592886491141503682831594979, 7.51504699865793258813499446877, 8.112061797823225156911589851846, 9.013599198798047570761593263412, 10.37302270266965707560680959513, 10.87391940823872711300127965834
