| L(s) = 1 | − 28·5-s + 98·7-s − 596·11-s + 532·13-s − 2.10e3·17-s + 2.74e3·19-s − 3.66e3·23-s − 1.55e3·25-s + 720·29-s − 3.97e3·31-s − 2.74e3·35-s + 6.78e3·37-s + 4.00e3·41-s − 1.94e4·43-s + 2.15e4·47-s + 7.20e3·49-s − 5.75e4·53-s + 1.66e4·55-s + 2.23e4·59-s − 3.87e4·61-s − 1.48e4·65-s − 7.28e3·67-s + 3.93e3·71-s + 1.92e4·73-s − 5.84e4·77-s − 1.56e4·79-s + 2.26e4·83-s + ⋯ |
| L(s) = 1 | − 0.500·5-s + 0.755·7-s − 1.48·11-s + 0.873·13-s − 1.76·17-s + 1.74·19-s − 1.44·23-s − 0.498·25-s + 0.158·29-s − 0.743·31-s − 0.378·35-s + 0.815·37-s + 0.371·41-s − 1.60·43-s + 1.42·47-s + 3/7·49-s − 2.81·53-s + 0.743·55-s + 0.835·59-s − 1.33·61-s − 0.437·65-s − 0.198·67-s + 0.0925·71-s + 0.423·73-s − 1.12·77-s − 0.281·79-s + 0.360·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3460717039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3460717039\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 28 T + 2342 T^{2} + 28 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 596 T + 209810 T^{2} + 596 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 532 T + 747678 T^{2} - 532 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2100 T + 3445630 T^{2} + 2100 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2744 T + 6243606 T^{2} - 2744 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3660 T + 14411722 T^{2} + 3660 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 720 T + 28281754 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3976 T + 26474190 T^{2} + 3976 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6788 T + 69768750 T^{2} - 6788 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4004 T + 203212622 T^{2} - 4004 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 19480 T + 337403910 T^{2} + 19480 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21560 T + 472446158 T^{2} - 21560 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 57576 T + 1664335546 T^{2} + 57576 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 22344 T + 1540856326 T^{2} - 22344 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 38724 T + 1471462046 T^{2} + 38724 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7288 T + 1671047286 T^{2} + 7288 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3932 T + 3486638858 T^{2} - 3932 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 19292 T + 2681901078 T^{2} - 19292 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15632 T + 5584565598 T^{2} + 15632 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22624 T + 7152673286 T^{2} - 22624 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 112812 T + 14123112334 T^{2} + 112812 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 36036 T + 8894455622 T^{2} + 36036 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.279505440311409510940030928037, −9.110013411779934129236227630656, −8.492119002216710172693424761220, −7.991562951016727310602571774333, −7.902885696039227995944884136849, −7.58644206470042750013546454121, −6.92497659327963682156438025707, −6.55628257758040627552209781223, −5.85434356974668533680820961808, −5.68361873837266505510381473814, −5.00374730944413479346763454658, −4.76886433875346986183380356998, −4.01237752265643661341368729524, −3.93076841298270700560895154842, −2.95253357705356951539059783139, −2.82220296945142636121080937299, −1.88676019199566580932694813641, −1.71116122489511082502060352037, −0.871090507384922915339981587272, −0.12879740626776010187684498022,
0.12879740626776010187684498022, 0.871090507384922915339981587272, 1.71116122489511082502060352037, 1.88676019199566580932694813641, 2.82220296945142636121080937299, 2.95253357705356951539059783139, 3.93076841298270700560895154842, 4.01237752265643661341368729524, 4.76886433875346986183380356998, 5.00374730944413479346763454658, 5.68361873837266505510381473814, 5.85434356974668533680820961808, 6.55628257758040627552209781223, 6.92497659327963682156438025707, 7.58644206470042750013546454121, 7.902885696039227995944884136849, 7.991562951016727310602571774333, 8.492119002216710172693424761220, 9.110013411779934129236227630656, 9.279505440311409510940030928037
