| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.0647 + 0.0747i)3-s + (−0.959 + 0.281i)4-s + (0.841 − 0.540i)5-s + (0.0832 + 0.0534i)6-s + (0.155 + 1.07i)7-s + (0.415 + 0.909i)8-s + (0.425 + 2.95i)9-s + (−0.654 − 0.755i)10-s + (−2.85 + 1.83i)11-s + (0.0410 − 0.0899i)12-s + (−0.933 + 2.04i)13-s + (1.04 − 0.307i)14-s + (−0.0140 + 0.0979i)15-s + (0.841 − 0.540i)16-s + (−1.70 − 0.501i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.0374 + 0.0431i)3-s + (−0.479 + 0.140i)4-s + (0.376 − 0.241i)5-s + (0.0339 + 0.0218i)6-s + (0.0586 + 0.408i)7-s + (0.146 + 0.321i)8-s + (0.141 + 0.986i)9-s + (−0.207 − 0.238i)10-s + (−0.861 + 0.553i)11-s + (0.0118 − 0.0259i)12-s + (−0.258 + 0.567i)13-s + (0.279 − 0.0821i)14-s + (−0.00363 + 0.0252i)15-s + (0.210 − 0.135i)16-s + (−0.414 − 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.961894 + 0.458668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.961894 + 0.458668i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.74 + 7.71i)T \) |
| good | 3 | \( 1 + (0.0647 - 0.0747i)T + (-0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.155 - 1.07i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (2.85 - 1.83i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.933 - 2.04i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (1.70 + 0.501i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.778 - 5.41i)T + (-18.2 - 5.35i)T^{2} \) |
| 23 | \( 1 + (0.951 - 1.09i)T + (-3.27 - 22.7i)T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + (-0.366 - 0.802i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 + (0.252 + 0.0742i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-8.40 - 2.46i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (4.52 - 5.22i)T + (-6.68 - 46.5i)T^{2} \) |
| 53 | \( 1 + (-7.35 + 2.15i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-3.67 - 8.04i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (4.48 + 2.88i)T + (25.3 + 55.4i)T^{2} \) |
| 71 | \( 1 + (4.72 - 1.38i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.19 - 5.26i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (4.17 - 9.13i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-13.4 + 8.63i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (9.79 + 11.2i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−10.52488446504652744128107423825, −9.943249143392945744625262676243, −9.065109469503260215304612983329, −8.125178903550711274144182022234, −7.35142496391693115071671898070, −5.90634302028552258420531582171, −5.03738804830345761747075698082, −4.16836589743104207467737576779, −2.58120051041348700693512227554, −1.78654597653908584297882816443,
0.58314588073880220962345735890, 2.63410000860670293115246904981, 3.91233813195794917892339563550, 5.09729130884207816760220652963, 6.01047271187573432309501751316, 6.84453570730609726827185951157, 7.63242315998560044616887280552, 8.650917392714719398160319525568, 9.400518778687614808259820495213, 10.34765029349856968379318408647
