| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−1.93 − 2.22i)3-s + (−0.959 − 0.281i)4-s + (−0.841 − 0.540i)5-s + (2.48 − 1.59i)6-s + (−0.342 + 2.38i)7-s + (0.415 − 0.909i)8-s + (−0.811 + 5.64i)9-s + (0.654 − 0.755i)10-s + (2.78 + 1.78i)11-s + (1.22 + 2.68i)12-s + (−0.155 − 0.339i)13-s + (−2.30 − 0.678i)14-s + (0.419 + 2.91i)15-s + (0.841 + 0.540i)16-s + (−0.901 + 0.264i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−1.11 − 1.28i)3-s + (−0.479 − 0.140i)4-s + (−0.376 − 0.241i)5-s + (1.01 − 0.651i)6-s + (−0.129 + 0.900i)7-s + (0.146 − 0.321i)8-s + (−0.270 + 1.88i)9-s + (0.207 − 0.238i)10-s + (0.839 + 0.539i)11-s + (0.353 + 0.774i)12-s + (−0.0430 − 0.0941i)13-s + (−0.617 − 0.181i)14-s + (0.108 + 0.753i)15-s + (0.210 + 0.135i)16-s + (−0.218 + 0.0641i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.787852 - 0.102417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.787852 - 0.102417i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (2.33 + 7.84i)T \) |
| good | 3 | \( 1 + (1.93 + 2.22i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (0.342 - 2.38i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.78 - 1.78i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.155 + 0.339i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.901 - 0.264i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.391 + 2.72i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (1.65 + 1.91i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 9.25T + 29T^{2} \) |
| 31 | \( 1 + (-2.58 + 5.65i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 - 4.40T + 37T^{2} \) |
| 41 | \( 1 + (-0.680 + 0.199i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-6.79 + 1.99i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (7.04 + 8.13i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-5.75 - 1.69i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (1.45 - 3.17i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.132 + 0.0849i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-15.4 - 4.54i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (2.83 - 1.82i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.29 + 2.83i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.57 - 1.01i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (1.05 - 1.22i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−10.60072813326685137737727608254, −9.403333610525072939082456046575, −8.486211977516613338840220672948, −7.67841673543317345598811070726, −6.71524255974168950706119143495, −6.27012554255261043922393537055, −5.31796894108152019647938770208, −4.36092363299007839006949707847, −2.31072980999714273245579838705, −0.77103494059517187601356221095,
0.888325380449743064393231281795, 3.22761872913447997923092350948, 4.08517215665719824763762570770, 4.67411403646335154140050141500, 5.93822722958601850202740745390, 6.78246444193935372361905637844, 8.148619109708599011799742663151, 9.219256959424560703705103104750, 10.03309026322781796084382408625, 10.55248491135477089554309018218
