L(s) = 1 | + (0.365 + 0.930i)3-s + (−0.222 + 0.974i)4-s + (−0.826 − 1.43i)7-s + (−0.733 + 0.680i)9-s + (−0.988 + 0.149i)12-s + (−1.21 − 0.825i)13-s + (−0.900 − 0.433i)16-s + (−0.535 − 0.496i)19-s + (1.03 − 1.29i)21-s + (−0.900 − 0.433i)27-s + (1.57 − 0.487i)28-s + (−1.23 + 0.185i)31-s + (−0.500 − 0.866i)36-s + (0.733 − 1.26i)37-s + (0.326 − 1.42i)39-s + ⋯ |
L(s) = 1 | + (0.365 + 0.930i)3-s + (−0.222 + 0.974i)4-s + (−0.826 − 1.43i)7-s + (−0.733 + 0.680i)9-s + (−0.988 + 0.149i)12-s + (−1.21 − 0.825i)13-s + (−0.900 − 0.433i)16-s + (−0.535 − 0.496i)19-s + (1.03 − 1.29i)21-s + (−0.900 − 0.433i)27-s + (1.57 − 0.487i)28-s + (−1.23 + 0.185i)31-s + (−0.500 − 0.866i)36-s + (0.733 − 1.26i)37-s + (0.326 − 1.42i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2540946751\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2540946751\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.365 - 0.930i)T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + (0.988 + 0.149i)T \) |
good | 2 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 7 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \) |
| 17 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 19 | \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \) |
| 23 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 29 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 31 | \( 1 + (1.23 - 0.185i)T + (0.955 - 0.294i)T^{2} \) |
| 37 | \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (1.88 + 0.284i)T + (0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2} \) |
| 71 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 73 | \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \) |
| 79 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 89 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \) |
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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
−8.745815388446470268726458038160, −7.71989383729605086723826702035, −7.45643730756484527182628954626, −6.56391728399530172363701850820, −5.33153054077347875683297254813, −4.53139674172197826800561918844, −3.87001290496902459102186385068, −3.24052874775070779987042174177, −2.43480537695740912490042634863, −0.13310392418466443197309554418,
1.70044618905347654427632892569, 2.32529533406502560222792597654, 3.24063221944469379532165150153, 4.57679477426743593539499442875, 5.45475060363901344075891301970, 6.16600916904065280138920058142, 6.60038886264678967342790006844, 7.48141564564268548074117127472, 8.455392797634714946121592550871, 9.114803328267802322255899700206