L(s) = 1 | − 1.93i·5-s + 1.73·13-s − 0.517i·17-s − 2.73·25-s − 0.517i·29-s + 37-s + 1.41i·41-s − 49-s − 1.41i·53-s − 61-s − 3.34i·65-s − 1.73·73-s − 0.999·85-s − 1.93i·89-s + 1.41i·101-s + ⋯ |
L(s) = 1 | − 1.93i·5-s + 1.73·13-s − 0.517i·17-s − 2.73·25-s − 0.517i·29-s + 37-s + 1.41i·41-s − 49-s − 1.41i·53-s − 61-s − 3.34i·65-s − 1.73·73-s − 0.999·85-s − 1.93i·89-s + 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.249637285\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.249637285\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.93iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 + 0.517iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.517iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.73T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.93iT - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.796926749919001309642402211323, −8.288710628327956165867583659676, −7.63840617599287154503483993983, −6.31351530272855177733033964083, −5.78204119813552837580542365884, −4.83431023790833045164887381187, −4.28920629871198502152498334337, −3.30131844448192072836828121224, −1.76529301808682896648500888353, −0.893359783102715216522950490409,
1.65982862005967675453417260060, 2.81354725631869922620911950990, 3.49481691729189560328837598482, 4.23001315107718891240405761203, 5.77274647662331976537431601963, 6.18149841198059351207588477153, 6.92002666290696829676584282065, 7.62249591586453300543186867180, 8.419215886236822913275994690681, 9.283687659218333133334932221320