| L(s) = 1 | + (1.71 − 0.386i)2-s + (−1.65 − 0.498i)3-s + (0.991 − 0.470i)4-s + (0.334 + 0.743i)5-s + (−3.04 − 0.215i)6-s + (−4.15 + 0.589i)7-s + (−1.25 + 0.979i)8-s + (2.50 + 1.65i)9-s + (0.862 + 1.14i)10-s + (−1.17 + 1.27i)11-s + (−1.87 + 0.285i)12-s + (0.0554 + 3.60i)13-s + (−6.90 + 2.61i)14-s + (−0.184 − 1.40i)15-s + (−3.15 + 3.86i)16-s + (2.61 + 1.96i)17-s + ⋯ |
| L(s) = 1 | + (1.21 − 0.273i)2-s + (−0.957 − 0.287i)3-s + (0.495 − 0.235i)4-s + (0.149 + 0.332i)5-s + (−1.24 − 0.0877i)6-s + (−1.57 + 0.222i)7-s + (−0.442 + 0.346i)8-s + (0.834 + 0.551i)9-s + (0.272 + 0.362i)10-s + (−0.355 + 0.385i)11-s + (−0.542 + 0.0824i)12-s + (0.0153 + 0.999i)13-s + (−1.84 + 0.699i)14-s + (−0.0475 − 0.361i)15-s + (−0.789 + 0.966i)16-s + (0.634 + 0.476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.514554 + 0.711105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.514554 + 0.711105i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 + 0.498i)T \) |
| 13 | \( 1 + (-0.0554 - 3.60i)T \) |
| good | 2 | \( 1 + (-1.71 + 0.386i)T + (1.80 - 0.857i)T^{2} \) |
| 5 | \( 1 + (-0.334 - 0.743i)T + (-3.31 + 3.74i)T^{2} \) |
| 7 | \( 1 + (4.15 - 0.589i)T + (6.72 - 1.94i)T^{2} \) |
| 11 | \( 1 + (1.17 - 1.27i)T + (-0.885 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-2.61 - 1.96i)T + (4.72 + 16.3i)T^{2} \) |
| 19 | \( 1 + (1.34 - 0.359i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (2.94 + 5.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.20 - 5.06i)T + (-12.4 - 26.2i)T^{2} \) |
| 31 | \( 1 + (-0.231 - 0.0139i)T + (30.7 + 3.73i)T^{2} \) |
| 37 | \( 1 + (2.75 - 1.37i)T + (22.2 - 29.5i)T^{2} \) |
| 41 | \( 1 + (8.48 + 0.170i)T + (40.9 + 1.65i)T^{2} \) |
| 43 | \( 1 + (-3.12 - 9.36i)T + (-34.3 + 25.8i)T^{2} \) |
| 47 | \( 1 + (-1.06 + 5.81i)T + (-43.9 - 16.6i)T^{2} \) |
| 53 | \( 1 + (1.30 + 0.159i)T + (51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (-1.02 + 10.1i)T + (-57.8 - 11.8i)T^{2} \) |
| 61 | \( 1 + (-12.2 - 5.20i)T + (42.2 + 43.9i)T^{2} \) |
| 67 | \( 1 + (6.41 - 2.28i)T + (51.8 - 42.3i)T^{2} \) |
| 71 | \( 1 + (-3.22 - 5.84i)T + (-37.9 + 60.0i)T^{2} \) |
| 73 | \( 1 + (6.26 - 1.95i)T + (60.0 - 41.4i)T^{2} \) |
| 79 | \( 1 + (-7.93 - 11.4i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (-7.17 + 4.33i)T + (38.5 - 73.4i)T^{2} \) |
| 89 | \( 1 + (-0.401 + 1.49i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (6.27 + 8.70i)T + (-30.7 + 92.0i)T^{2} \) |
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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
−11.46405609233728816723116267177, −10.44841047950122363355816751568, −9.770475716553962224550088715116, −8.517043966280035753717542971339, −6.87778673415027780926362205054, −6.43684566013819518684590323305, −5.58583387771707197917477013421, −4.53337894941066945172486841791, −3.49347004578964981935457981168, −2.24907784424635322797434154140,
0.37665245084854317263266208466, 3.17762126436713290257697717310, 3.89383378272610270176723045605, 5.25489740920042302028899216884, 5.69686631254859888103283107884, 6.53349839071205699885494557321, 7.48561351915938187153601288035, 9.189863053083870288329271193006, 9.867689874111223722995606490698, 10.65604186573758551123705279889
