| L(s) = 1 | + (−0.266 − 0.223i)2-s + (0.939 − 0.342i)3-s + (−0.326 − 1.85i)4-s + (−0.233 + 1.32i)5-s + (−0.326 − 0.118i)6-s + (0.326 + 0.565i)7-s + (−0.673 + 1.16i)8-s + (0.766 − 0.642i)9-s + (0.358 − 0.300i)10-s + (−1.97 + 3.41i)11-s + (−0.939 − 1.62i)12-s + (−1.59 − 0.579i)13-s + (0.0393 − 0.223i)14-s + (0.233 + 1.32i)15-s + (−3.09 + 1.12i)16-s + (−2.93 − 2.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.188 − 0.157i)2-s + (0.542 − 0.197i)3-s + (−0.163 − 0.925i)4-s + (−0.104 + 0.593i)5-s + (−0.133 − 0.0484i)6-s + (0.123 + 0.213i)7-s + (−0.238 + 0.412i)8-s + (0.255 − 0.214i)9-s + (0.113 − 0.0951i)10-s + (−0.594 + 1.02i)11-s + (−0.271 − 0.469i)12-s + (−0.441 − 0.160i)13-s + (0.0105 − 0.0596i)14-s + (0.0604 + 0.342i)15-s + (−0.773 + 0.281i)16-s + (−0.712 − 0.598i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.830899 - 0.212643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.830899 - 0.212643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (-2.82 - 3.31i)T \) |
| good | 2 | \( 1 + (0.266 + 0.223i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (0.233 - 1.32i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.326 - 0.565i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.97 - 3.41i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.59 + 0.579i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.93 + 2.46i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.631 + 3.58i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-8.05 + 6.76i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.23 + 5.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 41 | \( 1 + (-1.41 + 0.516i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.62 - 9.22i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (9.77 - 8.20i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (1.87 + 10.6i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (2.12 + 1.78i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.748 - 4.24i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.04 + 5.06i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.932 - 5.28i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (5.51 - 2.00i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.93 + 1.06i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.25 - 3.90i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.4 - 3.80i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.13 - 4.30i)T + (16.8 + 95.5i)T^{2} \) |
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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
−14.92765749725931058585307823019, −14.30789900657442908276407978796, −13.03957437735674739418629223730, −11.63102206191647303286248474243, −10.30601413190672236510563488076, −9.503380134016346215302701319297, −7.966260175545973169527337362354, −6.57763013030256104277547696419, −4.83644641260977243684267893874, −2.42377951463254828862478205074,
3.22988379104188754171975527228, 4.85626218196179768684971916420, 7.08700500627690786029183846298, 8.379149423129916485373255258390, 9.015600755718425727796735086501, 10.67894006662901917802332831170, 12.12036278568923649030298594489, 13.17702037319046946195127704938, 14.03548670747572568608899896912, 15.65056028042944147605242121623
