| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s + 2·11-s − 2·15-s + 2·17-s − 6·19-s − 8·21-s − 6·23-s + 25-s − 4·27-s − 2·29-s − 6·31-s + 4·33-s + 4·35-s − 2·37-s − 10·41-s − 10·43-s − 45-s − 12·47-s + 9·49-s + 4·51-s − 2·53-s − 2·55-s − 12·57-s − 10·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.516·15-s + 0.485·17-s − 1.37·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 0.371·29-s − 1.07·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s − 1.56·41-s − 1.52·43-s − 0.149·45-s − 1.75·47-s + 9/7·49-s + 0.560·51-s − 0.274·53-s − 0.269·55-s − 1.58·57-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4333389331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4333389331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.46179775750280, −13.96607331248861, −13.33862377377418, −13.07660737781026, −12.43105512712449, −12.06220383179070, −11.40906534837524, −10.77272240460992, −10.07253019894826, −9.718347475361168, −9.279406971019269, −8.635461217585392, −8.269168159703413, −7.774064518867777, −6.988224535305773, −6.560859401415517, −6.113079468512900, −5.335974087943003, −4.500088383758563, −3.711556043550149, −3.509459834893519, −3.045389512652406, −2.095616971686998, −1.668880779153596, −0.1956042675839830,
0.1956042675839830, 1.668880779153596, 2.095616971686998, 3.045389512652406, 3.509459834893519, 3.711556043550149, 4.500088383758563, 5.335974087943003, 6.113079468512900, 6.560859401415517, 6.988224535305773, 7.774064518867777, 8.269168159703413, 8.635461217585392, 9.279406971019269, 9.718347475361168, 10.07253019894826, 10.77272240460992, 11.40906534837524, 12.06220383179070, 12.43105512712449, 13.07660737781026, 13.33862377377418, 13.96607331248861, 14.46179775750280
