L(s) = 1 | + (−0.669 + 0.743i)3-s + (0.207 + 0.978i)4-s + (0.866 + 0.5i)5-s + (−0.104 − 0.994i)9-s + (0.994 − 0.104i)11-s + (−0.866 − 0.5i)12-s + (−0.951 + 0.309i)15-s + (−0.913 + 0.406i)16-s + (−0.309 + 0.951i)20-s + (0.638 + 1.66i)23-s + (0.499 + 0.866i)25-s + (0.809 + 0.587i)27-s + (0.544 − 0.604i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.0809 − 0.511i)37-s + ⋯ |
L(s) = 1 | + (−0.669 + 0.743i)3-s + (0.207 + 0.978i)4-s + (0.866 + 0.5i)5-s + (−0.104 − 0.994i)9-s + (0.994 − 0.104i)11-s + (−0.866 − 0.5i)12-s + (−0.951 + 0.309i)15-s + (−0.913 + 0.406i)16-s + (−0.309 + 0.951i)20-s + (0.638 + 1.66i)23-s + (0.499 + 0.866i)25-s + (0.809 + 0.587i)27-s + (0.544 − 0.604i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.0809 − 0.511i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.223401194\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.223401194\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.994 + 0.104i)T \) |
good | 2 | \( 1 + (-0.207 - 0.978i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (0.207 - 0.978i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.638 - 1.66i)T + (-0.743 + 0.669i)T^{2} \) |
| 29 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.544 + 0.604i)T + (-0.104 - 0.994i)T^{2} \) |
| 37 | \( 1 + (0.0809 + 0.511i)T + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (1.92 + 0.101i)T + (0.994 + 0.104i)T^{2} \) |
| 53 | \( 1 + (0.638 - 0.325i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (0.0218 - 0.207i)T + (-0.978 - 0.207i)T^{2} \) |
| 61 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 67 | \( 1 + (-0.104 + 0.00547i)T + (0.994 - 0.104i)T^{2} \) |
| 71 | \( 1 + (1.89 + 0.614i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (0.406 - 0.913i)T^{2} \) |
| 89 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.0977 + 1.86i)T + (-0.994 - 0.104i)T^{2} \) |
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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
−9.302211900644513133871483683461, −8.901693326684694280985128035636, −7.70509037224461838159821413705, −6.89719843882165746401736895991, −6.28857192794090035040065803274, −5.52355217977958608382156714100, −4.54423679010891864625942600948, −3.62456841396424541111100955912, −3.00525282146404794879266836912, −1.63558270108971702210706677406,
0.973120736973060952734456605098, 1.71673601408096533797655728523, 2.70690943402566951287275116680, 4.52508918585400459365150913393, 5.03407566793224243740411568060, 5.92680171899888287092827063669, 6.53297254098689565601263321422, 6.89245300266164986271816975663, 8.197339065252480891374048267157, 8.936122116301773316559876897690