| L(s) = 1 | + (0.757 − 0.652i)2-s + (0.301 + 0.159i)3-s + (0.149 − 0.988i)4-s + (−1.63 − 1.52i)5-s + (0.332 − 0.0759i)6-s + (−1.77 + 1.96i)7-s + (−0.532 − 0.846i)8-s + (−1.62 − 2.38i)9-s + (−2.23 − 0.0840i)10-s + (−3.79 − 2.58i)11-s + (0.202 − 0.274i)12-s + (0.264 + 0.0924i)13-s + (−0.0671 + 2.64i)14-s + (−0.251 − 0.720i)15-s + (−0.955 − 0.294i)16-s + (−1.31 − 3.02i)17-s + ⋯ |
| L(s) = 1 | + (0.535 − 0.461i)2-s + (0.174 + 0.0920i)3-s + (0.0745 − 0.494i)4-s + (−0.732 − 0.680i)5-s + (0.135 − 0.0310i)6-s + (−0.671 + 0.741i)7-s + (−0.188 − 0.299i)8-s + (−0.541 − 0.794i)9-s + (−0.706 − 0.0265i)10-s + (−1.14 − 0.780i)11-s + (0.0585 − 0.0792i)12-s + (0.0732 + 0.0256i)13-s + (−0.0179 + 0.706i)14-s + (−0.0650 − 0.186i)15-s + (−0.238 − 0.0736i)16-s + (−0.319 − 0.733i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.243882 - 1.00556i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.243882 - 1.00556i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.757 + 0.652i)T \) |
| 5 | \( 1 + (1.63 + 1.52i)T \) |
| 7 | \( 1 + (1.77 - 1.96i)T \) |
| good | 3 | \( 1 + (-0.301 - 0.159i)T + (1.68 + 2.47i)T^{2} \) |
| 11 | \( 1 + (3.79 + 2.58i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-0.264 - 0.0924i)T + (10.1 + 8.10i)T^{2} \) |
| 17 | \( 1 + (1.31 + 3.02i)T + (-11.5 + 12.4i)T^{2} \) |
| 19 | \( 1 + (-3.26 + 5.64i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.32 - 1.88i)T + (15.6 + 16.8i)T^{2} \) |
| 29 | \( 1 + (8.22 - 6.56i)T + (6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-7.11 + 4.10i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.42 - 3.26i)T + (10.9 + 35.3i)T^{2} \) |
| 41 | \( 1 + (0.656 + 0.149i)T + (36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.06 - 1.29i)T + (18.6 + 38.7i)T^{2} \) |
| 47 | \( 1 + (6.35 + 7.38i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-3.59 + 2.65i)T + (15.6 - 50.6i)T^{2} \) |
| 59 | \( 1 + (10.5 - 9.82i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.0589 - 0.391i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-1.32 + 4.92i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.25 + 7.84i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-9.59 + 11.1i)T + (-10.8 - 72.1i)T^{2} \) |
| 79 | \( 1 + (-4.22 - 2.43i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.798 - 2.28i)T + (-64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (-5.63 + 3.84i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (5.64 - 5.64i)T - 97iT^{2} \) |
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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
−10.92811964422552760867684089460, −9.396524788906355288441850051457, −9.106079692398817351896760924761, −8.003098766012123135528298487160, −6.77586926640129559628370453054, −5.57549078701037054350733984582, −4.89749834209526747809820118769, −3.40084674154714953606129350506, −2.80638352942192418504421682299, −0.49072574458657513852423262979,
2.53002929374800665270082888365, 3.56202689037301230754628326318, 4.59752004538832940242306880945, 5.80341041144902434051707413784, 6.84906519636155204160070581521, 7.76820845968143550856507050884, 8.073159193569383671205671317852, 9.700192130958180467768560466212, 10.59854691749440234493571995343, 11.23238279231337211005877727884
