L(s) = 1 | + (−0.0747 − 0.997i)7-s + (−0.255 + 0.531i)13-s + (1.72 − 0.997i)19-s + (0.0747 − 0.997i)25-s + (−0.258 − 0.149i)31-s + (0.722 + 0.108i)37-s + (0.367 − 1.61i)43-s + (−0.988 + 0.149i)49-s + (−0.233 + 1.54i)61-s + (0.733 − 1.26i)67-s + (−1.85 − 0.139i)73-s + (0.988 + 1.71i)79-s + (0.548 + 0.215i)91-s + 0.867i·97-s + (0.766 + 0.825i)103-s + ⋯ |
L(s) = 1 | + (−0.0747 − 0.997i)7-s + (−0.255 + 0.531i)13-s + (1.72 − 0.997i)19-s + (0.0747 − 0.997i)25-s + (−0.258 − 0.149i)31-s + (0.722 + 0.108i)37-s + (0.367 − 1.61i)43-s + (−0.988 + 0.149i)49-s + (−0.233 + 1.54i)61-s + (0.733 − 1.26i)67-s + (−1.85 − 0.139i)73-s + (0.988 + 1.71i)79-s + (0.548 + 0.215i)91-s + 0.867i·97-s + (0.766 + 0.825i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.124163738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.124163738\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.0747 + 0.997i)T \) |
good | 5 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 11 | \( 1 + (0.365 - 0.930i)T^{2} \) |
| 13 | \( 1 + (0.255 - 0.531i)T + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 19 | \( 1 + (-1.72 + 0.997i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.733 - 0.680i)T^{2} \) |
| 29 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.258 + 0.149i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.722 - 0.108i)T + (0.955 + 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (-0.367 + 1.61i)T + (-0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 53 | \( 1 + (0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 61 | \( 1 + (0.233 - 1.54i)T + (-0.955 - 0.294i)T^{2} \) |
| 67 | \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (1.85 + 0.139i)T + (0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 97 | \( 1 - 0.867iT - T^{2} \) |
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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
−9.470185540800695671751527048425, −8.672252717498051454385735806704, −7.58870851159184348582226007345, −7.17387875151104328610973349395, −6.30332874288383732027884341154, −5.22438810459413306858984561750, −4.42433657415649138768150582191, −3.53667959650754562735619996800, −2.44579602251257062697658870291, −0.961629043350915996328353933977,
1.47885762481960495065198131957, 2.78369725215534963855468984894, 3.51055965222060543619863560258, 4.84973508206990868060106129527, 5.56945834401888681853302687484, 6.20276963657999062571833586294, 7.40897830667634109878474398594, 7.917440745498468835517162740377, 8.889727218421243268873475873211, 9.570454062444755690158039712509