| L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.530 − 2.17i)5-s + (−0.309 − 0.951i)6-s − 0.273i·7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−2.06 + 0.847i)10-s + (3.87 − 2.81i)11-s + (−0.587 + 0.809i)12-s + (−1.01 + 1.40i)13-s + (−0.221 + 0.160i)14-s + (1.17 − 1.90i)15-s + (−0.809 − 0.587i)16-s + (−1.79 + 0.584i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (0.237 − 0.971i)5-s + (−0.126 − 0.388i)6-s − 0.103i·7-s + (0.336 − 0.109i)8-s + (0.269 + 0.195i)9-s + (−0.654 + 0.268i)10-s + (1.16 − 0.849i)11-s + (−0.169 + 0.233i)12-s + (−0.282 + 0.388i)13-s + (−0.0591 + 0.0429i)14-s + (0.303 − 0.491i)15-s + (−0.202 − 0.146i)16-s + (−0.436 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.982730 - 0.525114i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.982730 - 0.525114i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-0.530 + 2.17i)T \) |
| good | 7 | \( 1 + 0.273iT - 7T^{2} \) |
| 11 | \( 1 + (-3.87 + 2.81i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.01 - 1.40i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.79 - 0.584i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.930 - 2.86i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.73 + 3.75i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.80 - 8.64i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.18 - 3.64i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.29 - 7.28i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.13 - 4.45i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.88iT - 43T^{2} \) |
| 47 | \( 1 + (1.12 + 0.367i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (11.0 + 3.58i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (11.5 + 8.39i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.16 + 3.02i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.98 + 1.94i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.433 - 1.33i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.04 + 1.44i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.96 + 15.2i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-14.3 + 4.67i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (11.9 - 8.71i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.59 - 1.81i)T + (78.4 + 57.0i)T^{2} \) |
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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
−12.71320660009690125159979809836, −11.93067378429268151636429819703, −10.76715033414551324275350938544, −9.569100331369290307133255960002, −8.892301139856603270191867432224, −8.074379145642822118372552975666, −6.45551522219741679630543938894, −4.74200744321745670760205748768, −3.50128620220071149324657511328, −1.56973763982750033986901123134,
2.23124935863865476961635814660, 4.04057303394289109748380676273, 5.88737954755091702272459861193, 6.99989813486994549187912112795, 7.69910105947770053249847191713, 9.198978130257136573284568025919, 9.769520629704509800022818578351, 11.02203775510596198839719769297, 12.15365672067049536803065398072, 13.55513183774290759972772952251
