L(s) = 1 | + (0.0747 + 0.997i)2-s + (1.82 + 0.563i)3-s + (−0.988 + 0.149i)4-s + (−0.425 + 1.86i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (2.19 + 1.49i)9-s + (0.123 − 0.0841i)11-s + (−1.88 − 0.284i)12-s + (−0.134 − 0.0648i)13-s + (0.955 − 0.294i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−1.32 + 2.29i)18-s + (0.142 − 1.90i)21-s + (0.0931 + 0.116i)22-s + ⋯ |
L(s) = 1 | + (0.0747 + 0.997i)2-s + (1.82 + 0.563i)3-s + (−0.988 + 0.149i)4-s + (−0.425 + 1.86i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (2.19 + 1.49i)9-s + (0.123 − 0.0841i)11-s + (−1.88 − 0.284i)12-s + (−0.134 − 0.0648i)13-s + (0.955 − 0.294i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−1.32 + 2.29i)18-s + (0.142 − 1.90i)21-s + (0.0931 + 0.116i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.238794720\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.238794720\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0747 - 0.997i)T \) |
| 7 | \( 1 + (0.222 + 0.974i)T \) |
| 17 | \( 1 + (-0.365 - 0.930i)T \) |
good | 3 | \( 1 + (-1.82 - 0.563i)T + (0.826 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 11 | \( 1 + (-0.123 + 0.0841i)T + (0.365 - 0.930i)T^{2} \) |
| 13 | \( 1 + (0.134 + 0.0648i)T + (0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.658 - 1.67i)T + (-0.733 - 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 53 | \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 61 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.914 + 1.14i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-1.57 - 1.07i)T + (0.365 + 0.930i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.891608261987623576608509042534, −8.074213887203973433315587380315, −7.75495573670417432165360348978, −7.07959256962220705771968326601, −6.14609752127566784911163685219, −5.05671941706251804392176200881, −4.01861442956785365119411768031, −3.87409269413882433790114345419, −2.91577430461280319023609271984, −1.54217661353830773718073996063,
1.27134875628648076399096018637, 2.37980056555390516334230007232, 2.68801468291710407811873272939, 3.55779940686256011106135857630, 4.37508788872804051760674880803, 5.38193791748714577164553993903, 6.51831399897848205726747885479, 7.36744395939451979789925610647, 8.219469203684970407000506231368, 8.715766208756300307410375148581