| L(s) = 1 | − 2·3-s − 10·5-s − 32·9-s + 50·11-s + 26·13-s + 20·15-s − 88·17-s + 94·19-s + 122·23-s + 75·25-s + 82·27-s − 260·29-s + 402·31-s − 100·33-s + 320·37-s − 52·39-s − 536·41-s + 62·43-s + 320·45-s − 752·47-s − 2·49-s + 176·51-s + 248·53-s − 500·55-s − 188·57-s + 330·59-s + 620·61-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.894·5-s − 1.18·9-s + 1.37·11-s + 0.554·13-s + 0.344·15-s − 1.25·17-s + 1.13·19-s + 1.10·23-s + 3/5·25-s + 0.584·27-s − 1.66·29-s + 2.32·31-s − 0.527·33-s + 1.42·37-s − 0.213·39-s − 2.04·41-s + 0.219·43-s + 1.06·45-s − 2.33·47-s − 0.00583·49-s + 0.483·51-s + 0.642·53-s − 1.22·55-s − 0.436·57-s + 0.728·59-s + 1.30·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.374266707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.374266707\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 p^{2} T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 50 T + 2356 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 88 T + 8038 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 94 T + 11652 T^{2} - 94 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 122 T + 27884 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 260 T + 56482 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 402 T + 99964 T^{2} - 402 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 320 T + 107450 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 536 T + 187702 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 62 T + 9476 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 p T + 347122 T^{2} + 16 p^{4} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 248 T + 30334 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 330 T + 434772 T^{2} - 330 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 620 T + 467298 T^{2} - 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 460 T + 362282 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 734 T + 847300 T^{2} - 734 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 318994 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 924 T + 934966 T^{2} - 924 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2304 T + 2469994 T^{2} - 2304 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 748 T + 1481414 T^{2} + 748 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1924 T + 2107526 T^{2} - 1924 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.561655727451414422578052848078, −9.363974598071521814810302607259, −8.863426506261199491678828320997, −8.562741499608460030446327780293, −7.994059994445381748672648523544, −7.943223435862630348781104732529, −7.04315743914045807942826705673, −6.81701000439171087520847132177, −6.27911651976509954995888870535, −6.20784056751774781194425966964, −5.25855643673310660602734998646, −5.08685494934627497703097094142, −4.53884213442573757816532136734, −3.95167986049291729589579213864, −3.41599520106547917765647392955, −3.19908498008055885425174303580, −2.40102031607684155292545115403, −1.70598394909044382384323496201, −0.77608006343184836204666237835, −0.57799729189520997602507665185,
0.57799729189520997602507665185, 0.77608006343184836204666237835, 1.70598394909044382384323496201, 2.40102031607684155292545115403, 3.19908498008055885425174303580, 3.41599520106547917765647392955, 3.95167986049291729589579213864, 4.53884213442573757816532136734, 5.08685494934627497703097094142, 5.25855643673310660602734998646, 6.20784056751774781194425966964, 6.27911651976509954995888870535, 6.81701000439171087520847132177, 7.04315743914045807942826705673, 7.943223435862630348781104732529, 7.994059994445381748672648523544, 8.562741499608460030446327780293, 8.863426506261199491678828320997, 9.363974598071521814810302607259, 9.561655727451414422578052848078
