| L(s) = 1 | + (−1.39 − 2.71i)3-s + (−1.84 + 0.176i)5-s + (−1.47 + 2.19i)7-s + (−3.65 + 5.13i)9-s + (3.83 − 1.32i)11-s + (0.754 + 2.56i)13-s + (3.05 + 4.75i)15-s + (−2.87 − 1.15i)17-s + (−0.834 + 0.334i)19-s + (8.01 + 0.944i)21-s + (1.71 + 4.47i)23-s + (−1.53 + 0.295i)25-s + (9.96 + 1.43i)27-s + (0.0489 + 0.340i)29-s + (6.41 − 0.305i)31-s + ⋯ |
| L(s) = 1 | + (−0.806 − 1.56i)3-s + (−0.825 + 0.0788i)5-s + (−0.559 + 0.829i)7-s + (−1.21 + 1.71i)9-s + (1.15 − 0.400i)11-s + (0.209 + 0.712i)13-s + (0.789 + 1.22i)15-s + (−0.696 − 0.278i)17-s + (−0.191 + 0.0766i)19-s + (1.74 + 0.206i)21-s + (0.357 + 0.933i)23-s + (−0.306 + 0.0591i)25-s + (1.91 + 0.275i)27-s + (0.00908 + 0.0631i)29-s + (1.15 − 0.0548i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.634527 + 0.144544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.634527 + 0.144544i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.47 - 2.19i)T \) |
| 23 | \( 1 + (-1.71 - 4.47i)T \) |
| good | 3 | \( 1 + (1.39 + 2.71i)T + (-1.74 + 2.44i)T^{2} \) |
| 5 | \( 1 + (1.84 - 0.176i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-3.83 + 1.32i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-0.754 - 2.56i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (2.87 + 1.15i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (0.834 - 0.334i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.0489 - 0.340i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.41 + 0.305i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-8.89 - 6.33i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-0.0451 - 0.0206i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.464 + 0.723i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-0.203 + 0.117i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.25 + 3.41i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-5.56 - 1.34i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (6.26 + 3.22i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (-3.00 - 15.6i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-8.53 - 9.84i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (5.86 - 7.45i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (5.67 - 5.94i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.99 - 6.55i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.494 + 10.3i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (6.46 - 14.1i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.27786894513930099029556597547, −9.644793319198749428925285840531, −8.671200240556918178126875482382, −7.889948753047218940065574331045, −6.77092620087728715575538966181, −6.47675778240610900072232383234, −5.48216326624050256068249580623, −4.05645471629953903827024329278, −2.59884862867145185779810424132, −1.20340514473649598198601780640,
0.45559593318939806051706517986, 3.25411410379488821730744839922, 4.25165344820241275543605870138, 4.48403015772882397739791750742, 5.98511035866834054702995740934, 6.69755652869349040385399461423, 7.963711448761497673638738460429, 9.087948209088169996384375616216, 9.741130769277763902373713676720, 10.63354227406590850265018566575
