| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (−0.649 + 1.60i)3-s + (0.984 + 0.173i)4-s + (1.32 − 1.80i)5-s + (0.787 − 1.54i)6-s + (−2.99 − 2.09i)7-s + (−0.965 − 0.258i)8-s + (−2.15 − 2.08i)9-s + (−1.47 + 1.67i)10-s + (1.72 − 4.75i)11-s + (−0.918 + 1.46i)12-s + (0.119 + 1.36i)13-s + (2.80 + 2.35i)14-s + (2.02 + 3.29i)15-s + (0.939 + 0.342i)16-s + (2.52 − 0.677i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (−0.375 + 0.926i)3-s + (0.492 + 0.0868i)4-s + (0.592 − 0.805i)5-s + (0.321 − 0.629i)6-s + (−1.13 − 0.792i)7-s + (−0.341 − 0.0915i)8-s + (−0.718 − 0.695i)9-s + (−0.467 + 0.530i)10-s + (0.521 − 1.43i)11-s + (−0.265 + 0.423i)12-s + (0.0330 + 0.377i)13-s + (0.748 + 0.628i)14-s + (0.524 + 0.851i)15-s + (0.234 + 0.0855i)16-s + (0.612 − 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.580248 - 0.375745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.580248 - 0.375745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 + (0.649 - 1.60i)T \) |
| 5 | \( 1 + (-1.32 + 1.80i)T \) |
| good | 7 | \( 1 + (2.99 + 2.09i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.72 + 4.75i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.119 - 1.36i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-2.52 + 0.677i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.41 - 0.814i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.26 + 3.22i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-7.81 + 6.55i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.523 - 2.96i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.303 - 1.13i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.511 + 0.609i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.45 + 3.12i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (6.08 - 8.69i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.71 + 1.71i)T - 53iT^{2} \) |
| 59 | \( 1 + (12.1 - 4.42i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 6.45i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.38 + 0.471i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-12.3 - 7.13i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.898 + 3.35i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.03 + 9.57i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.797 + 9.11i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (7.31 + 12.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.03 - 1.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.57756740076590924081832067714, −10.44349635203258598154268810850, −9.913630888556788671998993055920, −9.068226563098430680805213343150, −8.277425117855454184844311040334, −6.49502504996882054134660563226, −5.93140600460805241497358594642, −4.37087436758026298460300154268, −3.17473209648906825435387984137, −0.69834864113795555580914593487,
1.86833098262892683378565506620, 3.03173586234011677113031732268, 5.45794693298745543835608772893, 6.49324983559461104816963748883, 6.94001134832997459430186997149, 8.125174955947789708865255030667, 9.445311669696562524101613805885, 10.01627967500592719661769435960, 11.09890980284314829820711055542, 12.26244644013809549245047174060
