L(s) = 1 | + (0.292 + 0.707i)3-s − i·4-s + (0.707 + 0.292i)7-s + (0.292 − 0.292i)9-s + (0.707 − 0.292i)12-s − 16-s − 17-s + 0.585i·21-s + (0.707 − 0.707i)25-s + (1.00 + 0.414i)27-s + (0.292 − 0.707i)28-s + (−0.292 − 0.292i)36-s + (0.707 + 1.70i)37-s + i·47-s + (−0.292 − 0.707i)48-s + (−0.292 − 0.292i)49-s + ⋯ |
L(s) = 1 | + (0.292 + 0.707i)3-s − i·4-s + (0.707 + 0.292i)7-s + (0.292 − 0.292i)9-s + (0.707 − 0.292i)12-s − 16-s − 17-s + 0.585i·21-s + (0.707 − 0.707i)25-s + (1.00 + 0.414i)27-s + (0.292 − 0.707i)28-s + (−0.292 − 0.292i)36-s + (0.707 + 1.70i)37-s + i·47-s + (−0.292 − 0.707i)48-s + (−0.292 − 0.292i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.144715778\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.144715778\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 47 | \( 1 - iT \) |
good | 2 | \( 1 + iT^{2} \) |
| 3 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 61 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
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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
−10.43359193067705147938585120747, −9.608944251214677514615236098992, −8.986359899023080206480232186033, −8.131894431859671650806072049790, −6.81267419863250867388740139920, −6.08019481742297483042924822901, −4.79876274307482899583782422644, −4.46965917503732683491478775816, −2.90150301558418022506540329796, −1.52141779790946051256389488615,
1.74253400638481136706800819564, 2.81681354362779972551415009849, 4.11512688321308378648967089916, 4.91331005623260117295370210204, 6.42239838291363031223512022359, 7.32592420002085790698621608933, 7.76807304728872810474613486327, 8.609641524747135311054981874582, 9.376179755129705291301232878869, 10.82126890142242602561803433222