| L(s) = 1 | + (−0.242 − 0.519i)2-s + (−0.193 + 1.72i)3-s + (1.07 − 1.28i)4-s + (−1.95 − 1.08i)5-s + (0.941 − 0.316i)6-s + (−0.984 − 3.67i)7-s + (−2.03 − 0.544i)8-s + (−2.92 − 0.665i)9-s + (−0.0903 + 1.27i)10-s + (−0.993 + 0.573i)11-s + (1.99 + 2.09i)12-s + (−1.00 + 0.700i)13-s + (−1.67 + 1.40i)14-s + (2.24 − 3.15i)15-s + (−0.370 − 2.10i)16-s + (−1.71 − 3.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.171 − 0.367i)2-s + (−0.111 + 0.993i)3-s + (0.537 − 0.640i)4-s + (−0.874 − 0.485i)5-s + (0.384 − 0.129i)6-s + (−0.372 − 1.38i)7-s + (−0.718 − 0.192i)8-s + (−0.975 − 0.221i)9-s + (−0.0285 + 0.404i)10-s + (−0.299 + 0.172i)11-s + (0.576 + 0.605i)12-s + (−0.277 + 0.194i)13-s + (−0.446 + 0.374i)14-s + (0.580 − 0.814i)15-s + (−0.0926 − 0.525i)16-s + (−0.415 − 0.890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.394498 - 0.651248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.394498 - 0.651248i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.193 - 1.72i)T \) |
| 5 | \( 1 + (1.95 + 1.08i)T \) |
| 19 | \( 1 + (-4.35 + 0.137i)T \) |
| good | 2 | \( 1 + (0.242 + 0.519i)T + (-1.28 + 1.53i)T^{2} \) |
| 7 | \( 1 + (0.984 + 3.67i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.993 - 0.573i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.00 - 0.700i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.71 + 3.67i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (2.47 - 0.216i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (-8.56 + 3.11i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.90 + 5.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.17 - 4.17i)T - 37iT^{2} \) |
| 41 | \( 1 + (6.21 - 1.09i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.83 + 0.510i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-6.79 - 3.17i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-11.6 + 1.01i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-0.160 - 0.0585i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.83 - 2.38i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.42 - 3.05i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (0.917 + 1.09i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-6.03 + 8.61i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (12.6 - 2.23i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.39 + 8.92i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.54 + 8.73i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 5.18i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.65724541186308837910938820441, −10.29441412115187302670118056249, −10.07619115424487097600553098148, −8.938418771008178564633126036629, −7.62573396570396888195125368737, −6.62710917763053051516862092218, −5.14341592946370581042261529106, −4.22245301483140074785677547008, −3.01554650592472646992304376534, −0.58694662933007946208308587915,
2.42871942013595258742068443325, 3.31483902010307269987797478664, 5.46754625183923640432994420659, 6.51232174764248142944126559941, 7.19621698833430817317780094884, 8.336059489047884877813752362749, 8.630520448014821521570989038185, 10.46042002118642639606890186201, 11.62303312723232559933636396973, 12.14001123528468240121420314985
