| L(s) = 1 | + (−0.459 + 0.413i)2-s + (−0.0646 + 0.614i)4-s + (−0.743 − 0.669i)5-s + (−0.309 + 0.535i)7-s + (−0.587 − 0.809i)8-s + 0.618·10-s + (0.743 − 0.669i)11-s + (−0.0794 − 0.373i)14-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.459 − 0.413i)20-s + (−0.0646 + 0.614i)22-s + (0.207 + 0.978i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.224i)28-s + (−0.658 − 1.47i)29-s + ⋯ |
| L(s) = 1 | + (−0.459 + 0.413i)2-s + (−0.0646 + 0.614i)4-s + (−0.743 − 0.669i)5-s + (−0.309 + 0.535i)7-s + (−0.587 − 0.809i)8-s + 0.618·10-s + (0.743 − 0.669i)11-s + (−0.0794 − 0.373i)14-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.459 − 0.413i)20-s + (−0.0646 + 0.614i)22-s + (0.207 + 0.978i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.224i)28-s + (−0.658 − 1.47i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7478213715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7478213715\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.743 + 0.669i)T \) |
| good | 2 | \( 1 + (0.459 - 0.413i)T + (0.104 - 0.994i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.743 + 0.669i)T + (0.104 - 0.994i)T^{2} \) |
| 13 | \( 1 + (-0.104 - 0.994i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \) |
| 29 | \( 1 + (0.658 + 1.47i)T + (-0.669 + 0.743i)T^{2} \) |
| 31 | \( 1 + (0.669 + 0.743i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.658 - 1.47i)T + (-0.669 + 0.743i)T^{2} \) |
| 53 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 61 | \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 67 | \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \) |
| 71 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (0.614 - 0.0646i)T + (0.978 - 0.207i)T^{2} \) |
| 89 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
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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
−9.309244999148642232511590786198, −8.640408559550704426187333559223, −7.916721242263277598640020760354, −7.45592589382914959748508356035, −6.35029170978153427157918742635, −5.70154971532837151137747251170, −4.47770103010308932400639739405, −3.67245820294498941245492625176, −2.92867718632348906129502072146, −1.08400043211464697062822940920,
0.816459609364607724552658043149, 2.17808236767767896473866573445, 3.29215037107903319855970567139, 4.12540859854794156419175221802, 5.17789684325368895422988463833, 6.09412751138016942186837549920, 7.15489814993628706706910706983, 7.37993170183307506425902439429, 8.639428462234207713911909929002, 9.310114014570906467371382051734
