L(s) = 1 | + (0.555 − 0.831i)2-s + (−0.382 − 0.923i)4-s + (0.636 + 1.53i)5-s + (−0.980 − 0.195i)8-s + (1.63 + 0.324i)10-s + (0.750 + 1.81i)11-s + (−0.923 − 0.382i)13-s + (−0.707 + 0.707i)16-s + (1.17 − 1.17i)20-s + (1.92 + 0.382i)22-s + (−1.24 + 1.24i)25-s + (−0.831 + 0.555i)26-s + (0.195 + 0.980i)32-s + (−0.324 − 1.63i)40-s + (−0.785 + 0.785i)41-s + ⋯ |
L(s) = 1 | + (0.555 − 0.831i)2-s + (−0.382 − 0.923i)4-s + (0.636 + 1.53i)5-s + (−0.980 − 0.195i)8-s + (1.63 + 0.324i)10-s + (0.750 + 1.81i)11-s + (−0.923 − 0.382i)13-s + (−0.707 + 0.707i)16-s + (1.17 − 1.17i)20-s + (1.92 + 0.382i)22-s + (−1.24 + 1.24i)25-s + (−0.831 + 0.555i)26-s + (0.195 + 0.980i)32-s + (−0.324 − 1.63i)40-s + (−0.785 + 0.785i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.675207259\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.675207259\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.555 + 0.831i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (0.923 + 0.382i)T \) |
good | 5 | \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.750 - 1.81i)T + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (0.785 - 0.785i)T - iT^{2} \) |
| 43 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 - 0.390iT - T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-1.02 + 0.425i)T + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + (-1.38 + 1.38i)T - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 0.765T + T^{2} \) |
| 83 | \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (1.38 + 1.38i)T + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.235783460402157193203617729895, −7.81014184721036776818757013941, −6.98315778236426012880611243198, −6.58283390413636991512588318245, −5.69435508251344172766450686251, −4.81965603873676187007636203876, −4.06443380209001536002401461883, −3.06339013521347888308082394934, −2.38571183089444794710302306711, −1.67403356825560934344185455718,
0.796706179152182334106899950194, 2.21619113403132724087536378468, 3.47573166319672344499301491135, 4.20685600749406647246244070242, 5.14875458997272617669113774674, 5.49593161558445825749012433219, 6.30569578785809617255997815439, 6.98793786549942548095393419289, 8.178680127359539785606655986534, 8.435827181950459531231193688656