L(s) = 1 | + (0.997 + 0.0697i)2-s + (1.31 + 1.26i)3-s + (0.990 + 0.139i)4-s + (−0.104 − 0.994i)5-s + (1.22 + 1.35i)6-s + (−0.939 − 1.62i)7-s + (0.978 + 0.207i)8-s + (0.0816 + 2.33i)9-s + (−0.0348 − 0.999i)10-s + (1.12 + 1.43i)12-s + (−0.823 − 1.68i)14-s + (1.12 − 1.43i)15-s + (0.961 + 0.275i)16-s + (−0.0816 + 2.33i)18-s + (0.0348 − 0.999i)20-s + (0.830 − 3.33i)21-s + ⋯ |
L(s) = 1 | + (0.997 + 0.0697i)2-s + (1.31 + 1.26i)3-s + (0.990 + 0.139i)4-s + (−0.104 − 0.994i)5-s + (1.22 + 1.35i)6-s + (−0.939 − 1.62i)7-s + (0.978 + 0.207i)8-s + (0.0816 + 2.33i)9-s + (−0.0348 − 0.999i)10-s + (1.12 + 1.43i)12-s + (−0.823 − 1.68i)14-s + (1.12 − 1.43i)15-s + (0.961 + 0.275i)16-s + (−0.0816 + 2.33i)18-s + (0.0348 − 0.999i)20-s + (0.830 − 3.33i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.443175439\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.443175439\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.997 - 0.0697i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 181 | \( 1 + (-0.990 - 0.139i)T \) |
good | 3 | \( 1 + (-1.31 - 1.26i)T + (0.0348 + 0.999i)T^{2} \) |
| 7 | \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.719 - 0.694i)T^{2} \) |
| 13 | \( 1 + (0.615 + 0.788i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.22 - 0.762i)T + (0.438 + 0.898i)T^{2} \) |
| 29 | \( 1 + (-0.473 - 0.100i)T + (0.913 + 0.406i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.882 + 0.469i)T^{2} \) |
| 41 | \( 1 + (1.35 - 0.719i)T + (0.559 - 0.829i)T^{2} \) |
| 43 | \( 1 + (0.194 + 1.10i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (1.20 + 1.54i)T + (-0.241 + 0.970i)T^{2} \) |
| 53 | \( 1 + (-0.559 - 0.829i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (1.35 - 1.13i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.139 - 0.155i)T + (-0.104 + 0.994i)T^{2} \) |
| 71 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 73 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.559 - 0.829i)T^{2} \) |
| 83 | \( 1 + (0.345 - 0.512i)T + (-0.374 - 0.927i)T^{2} \) |
| 89 | \( 1 + (-1.51 - 1.27i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (0.719 + 0.694i)T^{2} \) |
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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
−8.778266216799984473646385638044, −8.038199372641416430626970499931, −7.33740900235978505347535107543, −6.62485855000421499448044597513, −5.28738423679034701020610850372, −4.77802269258256443286399397753, −3.99030529857433174294986630776, −3.57234381519169367984301013591, −2.88935064293251140378519912403, −1.49684660252552202063979051502,
1.72048285867254289771098786150, 2.55644616569526024142254613020, 3.00976571258957344197381807783, 3.46915471714906855449650169907, 4.89448415933068152806579077832, 6.14485274021310380403226072173, 6.37863542205680953995177905881, 7.01462437866674469341157810506, 7.79896054905147050745732605521, 8.514699254963674409759519389104