| L(s) = 1 | − 10·4-s + 334·7-s − 470·13-s − 924·16-s + 2.72e3·19-s − 2.79e3·25-s − 3.34e3·28-s + 7.00e3·31-s + 2.62e4·37-s + 208·43-s + 5.00e4·49-s + 4.70e3·52-s − 1.47e4·61-s + 1.94e4·64-s + 7.77e4·67-s + 1.09e4·73-s − 2.72e4·76-s − 1.65e5·79-s − 1.56e5·91-s − 9.92e4·97-s + 2.79e4·100-s − 1.89e5·103-s + 2.49e5·109-s − 3.08e5·112-s + 2.61e5·121-s − 7.00e4·124-s + 127-s + ⋯ |
| L(s) = 1 | − 0.312·4-s + 2.57·7-s − 0.771·13-s − 0.902·16-s + 1.72·19-s − 0.894·25-s − 0.805·28-s + 1.30·31-s + 3.14·37-s + 0.0171·43-s + 2.97·49-s + 0.241·52-s − 0.508·61-s + 0.594·64-s + 2.11·67-s + 0.240·73-s − 0.540·76-s − 2.98·79-s − 1.98·91-s − 1.07·97-s + 0.279·100-s − 1.76·103-s + 2.01·109-s − 2.32·112-s + 1.62·121-s − 0.408·124-s + 5.50e−6·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.266168484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266168484\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 5 p T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2794 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 167 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 261962 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 235 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2808610 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1361 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 7063150 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40801114 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3500 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 13115 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 143238802 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 104 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 37745758 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 835271242 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 484532374 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7393 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 38861 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 3602362318 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 5465 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 82903 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7701562630 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3109663474 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 49603 T + p^{5} T^{2} )^{2} \) |
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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
−16.74430650279624425242780533139, −15.91368349828083401634052292965, −15.26032236392759854032447998901, −14.66070449210053976603178890432, −14.10703954616850405370163294143, −13.80117634488285171658816933005, −12.89415706528061783232935454164, −11.84419069072754654686991812163, −11.45355905347144470119494466548, −11.16137035077212669991273612776, −9.935885119629252332146229310373, −9.409305872844414406617692134725, −8.274946502929589927372230651342, −7.930641282386248022984232602830, −7.19328512718085639591101551867, −5.74527203674604307646717345609, −4.83916291033150074860993440614, −4.38282761481621820085750084826, −2.44692543196522632817817329616, −1.15606561193358013393957316331,
1.15606561193358013393957316331, 2.44692543196522632817817329616, 4.38282761481621820085750084826, 4.83916291033150074860993440614, 5.74527203674604307646717345609, 7.19328512718085639591101551867, 7.930641282386248022984232602830, 8.274946502929589927372230651342, 9.409305872844414406617692134725, 9.935885119629252332146229310373, 11.16137035077212669991273612776, 11.45355905347144470119494466548, 11.84419069072754654686991812163, 12.89415706528061783232935454164, 13.80117634488285171658816933005, 14.10703954616850405370163294143, 14.66070449210053976603178890432, 15.26032236392759854032447998901, 15.91368349828083401634052292965, 16.74430650279624425242780533139
