L(s) = 1 | + (0.707 − 1.70i)5-s − 1.41i·13-s + i·17-s + (−1.70 − 1.70i)25-s + (0.292 − 0.707i)29-s + (0.707 + 0.292i)37-s + (0.292 + 0.707i)41-s + (−0.707 + 0.707i)49-s + (−1 + i)53-s + (−0.707 − 1.70i)61-s + (−2.41 − 1.00i)65-s + (0.292 − 0.707i)73-s + (1.70 + 0.707i)85-s − 1.41i·89-s + (−0.707 + 1.70i)97-s + ⋯ |
L(s) = 1 | + (0.707 − 1.70i)5-s − 1.41i·13-s + i·17-s + (−1.70 − 1.70i)25-s + (0.292 − 0.707i)29-s + (0.707 + 0.292i)37-s + (0.292 + 0.707i)41-s + (−0.707 + 0.707i)49-s + (−1 + i)53-s + (−0.707 − 1.70i)61-s + (−2.41 − 1.00i)65-s + (0.292 − 0.707i)73-s + (1.70 + 0.707i)85-s − 1.41i·89-s + (−0.707 + 1.70i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.307652966\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.307652966\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - iT \) |
good | 5 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
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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
−8.973445375018758697439802915444, −8.060478811149004698591937420730, −7.888683795123313151272462205367, −6.25490272147500911839659640002, −5.86203553782364479241778180497, −4.96124705602761381212492358714, −4.38561259870922964929869643830, −3.16600916129079813808798032797, −1.91235442680749234644166954004, −0.902374834681912290179659613410,
1.81020485108707554383956443930, 2.63774307725563441118197266478, 3.45133258893759533085060405930, 4.51018055704137215229199077185, 5.57771346058493760431955655222, 6.40125741945701451919539954377, 6.96434781874073554109776476932, 7.45731888294496748424188035062, 8.669185716516377298881916053957, 9.521459482000262600670892639988