| L(s) = 1 | + (−1.98 + 0.404i)2-s + (−0.359 − 0.0584i)3-s + (1.92 − 0.819i)4-s + (0.199 + 0.809i)5-s + (0.735 − 0.0296i)6-s + (0.512 + 0.243i)7-s + (−0.149 + 0.102i)8-s + (−2.71 − 0.908i)9-s + (−0.722 − 1.52i)10-s + (1.27 + 3.83i)11-s + (−0.738 + 0.182i)12-s + (2.71 + 2.37i)13-s + (−1.11 − 0.274i)14-s + (−0.0244 − 0.302i)15-s + (−2.64 + 2.74i)16-s + (−3.39 + 7.15i)17-s + ⋯ |
| L(s) = 1 | + (−1.40 + 0.286i)2-s + (−0.207 − 0.0337i)3-s + (0.961 − 0.409i)4-s + (0.0892 + 0.362i)5-s + (0.300 − 0.0121i)6-s + (0.193 + 0.0919i)7-s + (−0.0527 + 0.0363i)8-s + (−0.906 − 0.302i)9-s + (−0.228 − 0.481i)10-s + (0.385 + 1.15i)11-s + (−0.213 + 0.0525i)12-s + (0.753 + 0.657i)13-s + (−0.297 − 0.0734i)14-s + (−0.00630 − 0.0781i)15-s + (−0.660 + 0.687i)16-s + (−0.822 + 1.73i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0987 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0987 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.319075 + 0.352290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.319075 + 0.352290i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.71 - 2.37i)T \) |
| good | 2 | \( 1 + (1.98 - 0.404i)T + (1.83 - 0.783i)T^{2} \) |
| 3 | \( 1 + (0.359 + 0.0584i)T + (2.84 + 0.950i)T^{2} \) |
| 5 | \( 1 + (-0.199 - 0.809i)T + (-4.42 + 2.32i)T^{2} \) |
| 7 | \( 1 + (-0.512 - 0.243i)T + (4.42 + 5.42i)T^{2} \) |
| 11 | \( 1 + (-1.27 - 3.83i)T + (-8.79 + 6.60i)T^{2} \) |
| 17 | \( 1 + (3.39 - 7.15i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-1.11 + 0.646i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.79 - 6.57i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.254 + 1.24i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 2.46i)T + (-17.6 - 25.5i)T^{2} \) |
| 37 | \( 1 + (-0.238 - 0.377i)T + (-15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.704 + 4.33i)T + (-38.8 - 12.9i)T^{2} \) |
| 43 | \( 1 + (-6.85 - 4.33i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-2.97 + 0.361i)T + (45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (4.71 + 6.83i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-3.83 + 3.68i)T + (2.37 - 58.9i)T^{2} \) |
| 61 | \( 1 + (6.56 + 0.529i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (3.93 - 9.23i)T + (-46.4 - 48.3i)T^{2} \) |
| 71 | \( 1 + (-10.8 - 8.81i)T + (14.2 + 69.5i)T^{2} \) |
| 73 | \( 1 + (3.93 - 4.44i)T + (-8.79 - 72.4i)T^{2} \) |
| 79 | \( 1 + (0.977 + 8.04i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-5.62 - 2.13i)T + (62.1 + 55.0i)T^{2} \) |
| 89 | \( 1 + (6.62 + 3.82i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.36 + 2.71i)T + (81.9 - 51.8i)T^{2} \) |
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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
−12.94912418187831044532137792900, −11.64878133612749930462367555836, −10.89377200767181369633523038199, −9.854761557608086393888896230694, −8.949551861463315254533998368212, −8.123024254627270289630916223944, −6.89962567596070211368179464955, −6.05028327354641079305105277076, −4.09622380000040505136747343512, −1.83031685246236075381973938398,
0.74216460973868753811302896239, 2.80671790630805568059989674683, 4.93214955910595350653913733995, 6.27722255396141827102257626034, 7.77101942651967379196368176479, 8.681785279800322952354124049196, 9.208976689419296337828785379780, 10.70157006076968059354669534841, 11.08121287135571253950049463177, 12.09640687828717514944093107484
