L(s) = 1 | + (−0.955 − 0.294i)2-s + (−0.680 + 1.73i)3-s + (0.826 + 0.563i)4-s + (1.16 − 1.45i)6-s + (−0.781 + 0.623i)7-s + (−0.623 − 0.781i)8-s + (−1.80 − 1.67i)9-s + (−0.432 + 0.400i)11-s + (−1.53 + 1.04i)12-s + (0.425 + 1.86i)13-s + (0.930 − 0.365i)14-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (1.23 + 2.13i)18-s + (−0.548 − 1.77i)21-s + (0.531 − 0.255i)22-s + ⋯ |
L(s) = 1 | + (−0.955 − 0.294i)2-s + (−0.680 + 1.73i)3-s + (0.826 + 0.563i)4-s + (1.16 − 1.45i)6-s + (−0.781 + 0.623i)7-s + (−0.623 − 0.781i)8-s + (−1.80 − 1.67i)9-s + (−0.432 + 0.400i)11-s + (−1.53 + 1.04i)12-s + (0.425 + 1.86i)13-s + (0.930 − 0.365i)14-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (1.23 + 2.13i)18-s + (−0.548 − 1.77i)21-s + (0.531 − 0.255i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4230543702\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4230543702\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.955 + 0.294i)T \) |
| 7 | \( 1 + (0.781 - 0.623i)T \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T \) |
good | 3 | \( 1 + (0.680 - 1.73i)T + (-0.733 - 0.680i)T^{2} \) |
| 5 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 11 | \( 1 + (0.432 - 0.400i)T + (0.0747 - 0.997i)T^{2} \) |
| 13 | \( 1 + (-0.425 - 1.86i)T + (-0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.145 - 1.94i)T + (-0.988 - 0.149i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.781 - 1.35i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 53 | \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 61 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.268 + 0.129i)T + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 79 | \( 1 + (-0.680 + 1.17i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.535 - 0.496i)T + (0.0747 + 0.997i)T^{2} \) |
| 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
−9.326258080441165893110557405239, −8.949471433196710333813880849562, −8.142341410820305595842093582201, −6.79208929937042037806693690105, −6.35714123667791602897867017405, −5.48612304857675647512506754350, −4.56071996690779469970604172663, −3.67857328106320680022660249644, −3.09338640828374147787118999156, −1.73946102456305619138633222701,
0.47456260015251222676412502069, 0.969150968131458083727713899063, 2.54070694411978905098580204868, 2.98569468219038885480339049238, 4.95676512041816027529702951235, 5.80794559920882278711925542433, 6.33597643708578859228812208938, 6.89058780995390188991204613401, 7.74798283267776786327087067249, 7.987888046473916228717754805081