| L(s) = 1 | + (−0.552 + 2.60i)2-s + (−0.533 − 0.592i)3-s + (−4.63 − 2.06i)4-s + 1.27i·5-s + (1.83 − 1.05i)6-s + (1.08 − 2.44i)7-s + (4.79 − 6.60i)8-s + (0.247 − 2.35i)9-s + (−3.31 − 0.705i)10-s + (3.49 − 0.367i)11-s + (1.24 + 3.84i)12-s + (1.56 + 3.24i)13-s + (5.76 + 4.18i)14-s + (0.755 − 0.680i)15-s + (7.73 + 8.58i)16-s + (0.199 − 1.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.390 + 1.83i)2-s + (−0.307 − 0.341i)3-s + (−2.31 − 1.03i)4-s + 0.570i·5-s + (0.748 − 0.432i)6-s + (0.411 − 0.924i)7-s + (1.69 − 2.33i)8-s + (0.0824 − 0.784i)9-s + (−1.04 − 0.223i)10-s + (1.05 − 0.110i)11-s + (0.360 + 1.10i)12-s + (0.435 + 0.900i)13-s + (1.53 + 1.11i)14-s + (0.195 − 0.175i)15-s + (1.93 + 2.14i)16-s + (0.0483 − 0.459i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.786482 + 0.513514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.786482 + 0.513514i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.56 - 3.24i)T \) |
| 31 | \( 1 + (0.591 - 5.53i)T \) |
| good | 2 | \( 1 + (0.552 - 2.60i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (0.533 + 0.592i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 - 1.27iT - 5T^{2} \) |
| 7 | \( 1 + (-1.08 + 2.44i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-3.49 + 0.367i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (-0.199 + 1.89i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (5.36 + 4.83i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.11 + 0.496i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-7.55 - 1.60i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (-1.22 - 0.708i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.94 + 9.15i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-3.12 + 3.46i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (6.40 + 2.08i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-8.26 - 6.00i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.28 + 10.7i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.704 - 1.22i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.08 - 2.35i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.68 - 1.01i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (2.95 + 4.06i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (10.4 + 7.59i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.06 + 2.62i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (10.6 - 1.11i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (0.586 - 1.31i)T + (-64.9 - 72.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.24983710487614074256658250377, −10.33461641929068026281812123083, −9.086497791050934372630054001946, −8.677005151440161675601513951008, −7.22786116506868166699983662744, −6.77952990138149263635779055297, −6.32575514975460300857027337542, −4.82975930745917460096146141458, −3.91768435416488764647814427688, −0.915101634815975243073597297176,
1.33032035981654550525626217324, 2.51964510451631339047015518662, 3.98471085950971697516915676645, 4.78343852998044844952874649538, 5.92719523516579561661296998803, 8.235276271061812203046675288118, 8.453266971660057483345474923839, 9.578463141518550367286382840619, 10.32914224054369365768338241574, 11.12524017741616685980285786872
