| L(s) = 1 | + (−2.07 − 0.923i)2-s + (0.0811 − 0.771i)3-s + (2.11 + 2.34i)4-s + 2.89·5-s + (−0.881 + 1.52i)6-s + (−1.22 − 1.35i)7-s + (−0.811 − 2.49i)8-s + (2.34 + 0.498i)9-s + (−6.00 − 2.67i)10-s + (−0.592 + 0.125i)11-s + (1.98 − 1.43i)12-s + (2.79 + 2.28i)13-s + (1.28 + 3.94i)14-s + (0.235 − 2.23i)15-s + (0.0367 − 0.349i)16-s + (3.62 + 0.771i)17-s + ⋯ |
| L(s) = 1 | + (−1.46 − 0.653i)2-s + (0.0468 − 0.445i)3-s + (1.05 + 1.17i)4-s + 1.29·5-s + (−0.359 + 0.623i)6-s + (−0.462 − 0.513i)7-s + (−0.286 − 0.882i)8-s + (0.781 + 0.166i)9-s + (−1.90 − 0.846i)10-s + (−0.178 + 0.0379i)11-s + (0.571 − 0.415i)12-s + (0.774 + 0.632i)13-s + (0.342 + 1.05i)14-s + (0.0606 − 0.577i)15-s + (0.00918 − 0.0874i)16-s + (0.880 + 0.187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.787541 - 0.431154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.787541 - 0.431154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.79 - 2.28i)T \) |
| 31 | \( 1 + (-3.60 + 4.24i)T \) |
| good | 2 | \( 1 + (2.07 + 0.923i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.0811 + 0.771i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 - 2.89T + 5T^{2} \) |
| 7 | \( 1 + (1.22 + 1.35i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (0.592 - 0.125i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (-3.62 - 0.771i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.537 - 5.11i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (4.24 - 4.71i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-8.09 - 3.60i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (5.91 + 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.274 + 0.122i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.976 + 9.29i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (2.48 + 1.80i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.363 + 1.11i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (7.63 - 3.39i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (0.326 - 0.565i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.15 - 5.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.51 + 0.321i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-2.72 + 8.39i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.97 - 15.3i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.27 - 0.929i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (2.91 - 0.620i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (11.1 + 12.3i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.59693051846477861637172603179, −10.09684417900318722642776194084, −9.608869086308326344721610698304, −8.533235576844100748324585949776, −7.61101485470197810418395500065, −6.69259437501381938119005810892, −5.62022852843818205182015310151, −3.69415101308741144940108345646, −2.07554381865622714923211165843, −1.29336349447706823968227287459,
1.25587235687445242438065303896, 2.90182965286703169896146269187, 4.87451135203750863820335962627, 6.18800164310301035335774929489, 6.57892645488500438430578475451, 7.982997215714716971365535978960, 8.777447339075137806713426419441, 9.704782933623820293272721560348, 10.02773042109792362154086311341, 10.74069226402304277627555714917
