| L(s) = 1 | + 16·7-s − 9·9-s + 36·17-s + 144·23-s − 74·25-s + 32·31-s − 180·41-s − 864·47-s − 494·49-s − 144·63-s − 720·71-s − 52·73-s − 1.02e3·79-s + 81·81-s + 1.26e3·89-s − 2.10e3·97-s + 16·103-s − 2.26e3·113-s + 576·119-s + 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 324·153-s + ⋯ |
| L(s) = 1 | + 0.863·7-s − 1/3·9-s + 0.513·17-s + 1.30·23-s − 0.591·25-s + 0.185·31-s − 0.685·41-s − 2.68·47-s − 1.44·49-s − 0.287·63-s − 1.20·71-s − 0.0833·73-s − 1.45·79-s + 1/9·81-s + 1.50·89-s − 2.20·97-s + 0.0153·103-s − 1.88·113-s + 0.443·119-s + 1.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.171·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.863911205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.863911205\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 74 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1366 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4294 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3718 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 5978 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 45290 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 57098 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 275878 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 491302 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 512 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 267770 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1054 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14858071133076545281746931845, −9.739919847427455563574475338619, −9.271255407467152516286926728112, −8.878101281256257653792667858166, −8.247958534128808148070462846459, −8.132352628961608129278890661530, −7.69316273785688947708914519294, −7.05880997640315337451379325834, −6.72748946652048656650596668711, −6.17970192409433000064031607951, −5.67790826027805101546871684544, −5.06331162241425501708168017139, −4.90002723888209589110040794150, −4.31241149195333604815098441241, −3.57826537335481845767631838800, −3.11615049532934010268840929431, −2.60245583633063442516468550735, −1.57668740369942432676037242689, −1.47459208676132857585444529943, −0.36974274430274173369867956676,
0.36974274430274173369867956676, 1.47459208676132857585444529943, 1.57668740369942432676037242689, 2.60245583633063442516468550735, 3.11615049532934010268840929431, 3.57826537335481845767631838800, 4.31241149195333604815098441241, 4.90002723888209589110040794150, 5.06331162241425501708168017139, 5.67790826027805101546871684544, 6.17970192409433000064031607951, 6.72748946652048656650596668711, 7.05880997640315337451379325834, 7.69316273785688947708914519294, 8.132352628961608129278890661530, 8.247958534128808148070462846459, 8.878101281256257653792667858166, 9.271255407467152516286926728112, 9.739919847427455563574475338619, 10.14858071133076545281746931845
