| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (−1.97 − 0.419i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.104 + 0.994i)10-s + (1.07 − 1.19i)11-s + (0.913 − 0.406i)12-s + (4.56 + 2.03i)13-s + (−1.35 − 1.49i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (2.39 + 2.66i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.0603 − 0.574i)3-s + (0.154 + 0.475i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (−0.746 − 0.158i)7-s + (−0.109 + 0.336i)8-s + (−0.326 + 0.0693i)9-s + (−0.0330 + 0.314i)10-s + (0.325 − 0.361i)11-s + (0.263 − 0.117i)12-s + (1.26 + 0.563i)13-s + (−0.360 − 0.400i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.581 + 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08730 + 0.707967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08730 + 0.707967i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.727 - 5.51i)T \) |
| good | 7 | \( 1 + (1.97 + 0.419i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-1.07 + 1.19i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.56 - 2.03i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.39 - 2.66i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.53 + 2.01i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.259 + 0.798i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.84 - 4.24i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (4.38 - 7.58i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.359 + 3.41i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-10.9 + 4.86i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (0.519 - 0.377i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (6.71 - 1.42i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.965 + 9.19i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 + (0.692 + 1.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.17 - 1.95i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-8.60 + 9.55i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (9.38 + 10.4i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (1.48 - 14.1i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (4.49 + 13.8i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.93 - 9.02i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.27548149912620570727724165219, −9.116760229513987607047862137755, −8.419133997713083296438631791735, −7.34647712454667180882813142860, −6.54311861670969540897051193839, −6.13764020404213581986193734653, −5.04722411164401423316505896989, −3.65510934504286422476057317914, −3.02028521227441995087456459504, −1.35786255643321657613461791979,
1.05987975923235544905868676040, 2.75594533608092174761548342449, 3.63025581868674069156581859033, 4.53873233056794114818766610083, 5.69025319630975486617257440400, 6.07906118580900713397254886673, 7.39374243170685829036705928447, 8.494528311229290201726716691017, 9.557529291895299132536007628369, 9.814935414204114986471005967920
