L(s) = 1 | + 2.14·2-s + (−0.301 − 0.521i)3-s + 2.58·4-s + (0.249 − 0.432i)5-s + (−0.644 − 1.11i)6-s + (2.01 + 3.48i)7-s + 1.25·8-s + (1.31 − 2.28i)9-s + (0.534 − 0.926i)10-s + (1.14 − 1.99i)11-s + (−0.778 − 1.34i)12-s + (−0.5 + 0.866i)13-s + (4.31 + 7.47i)14-s − 0.300·15-s − 2.48·16-s + (0.938 + 1.62i)17-s + ⋯ |
L(s) = 1 | + 1.51·2-s + (−0.173 − 0.301i)3-s + 1.29·4-s + (0.111 − 0.193i)5-s + (−0.263 − 0.455i)6-s + (0.761 + 1.31i)7-s + 0.442·8-s + (0.439 − 0.761i)9-s + (0.169 − 0.292i)10-s + (0.346 − 0.600i)11-s + (−0.224 − 0.389i)12-s + (−0.138 + 0.240i)13-s + (1.15 + 1.99i)14-s − 0.0776·15-s − 0.622·16-s + (0.227 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.99774 - 0.269360i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.99774 - 0.269360i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (4.97 - 2.50i)T \) |
good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 + (0.301 + 0.521i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.249 + 0.432i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.01 - 3.48i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.14 + 1.99i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.938 - 1.62i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.05 + 3.55i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 37 | \( 1 + (4.34 + 7.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.02 - 8.69i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.16 + 8.94i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + (4.26 - 7.39i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.60 - 9.70i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + (-3.45 + 5.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.69 - 9.86i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.64 + 9.78i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.487 - 0.843i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.735 + 1.27i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 0.499T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−11.80269439180540497978264352175, −10.77577722762718655302321259112, −9.196698171625844710835882317392, −8.590802728107973615571453847609, −7.06204628561408528468449368928, −6.12629182639512999958862062949, −5.43374759039960228380525967693, −4.44905101083912471853788736537, −3.25498430113846454166743025538, −1.88763374754740957359149658232,
1.95247799464558053170035903466, 3.61566751174099258196151741607, 4.50281017210421645336176932005, 5.05023511494275241390853178346, 6.38351067440648785418223242145, 7.27051312981983816776536559982, 8.225110745629715983581773561797, 9.992550319382144839901899598578, 10.49450354381207121755903278183, 11.45817013223198799990308347715