| L(s) = 1 | − 2·3-s − 14·7-s + 2·9-s + 8·11-s + 24·13-s + 40·17-s + 28·21-s − 18·23-s − 18·27-s + 80·31-s − 16·33-s − 32·37-s − 48·39-s + 64·41-s − 14·43-s + 62·47-s + 98·49-s − 80·51-s − 104·53-s − 28·63-s + 162·67-s + 36·69-s + 224·71-s + 88·73-s − 112·77-s − 13·81-s + 98·83-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 2·7-s + 2/9·9-s + 8/11·11-s + 1.84·13-s + 2.35·17-s + 4/3·21-s − 0.782·23-s − 2/3·27-s + 2.58·31-s − 0.484·33-s − 0.864·37-s − 1.23·39-s + 1.56·41-s − 0.325·43-s + 1.31·47-s + 2·49-s − 1.56·51-s − 1.96·53-s − 4/9·63-s + 2.41·67-s + 0.521·69-s + 3.15·71-s + 1.20·73-s − 1.45·77-s − 0.160·81-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.709274177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709274177\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 40 T + 800 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 62 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 526 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 32 T + 512 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 62 T + 1922 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 104 T + 5408 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5026 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 162 T + 13122 T^{2} - 162 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 112 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 88 T + 3872 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7298 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6238 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 88 T + 3872 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.21094356727767450999692234476, −10.81321270682472866927281974235, −10.41734914639437599321239868327, −9.722647143018227551827668992716, −9.580968173140962867665442901502, −9.391009028096546532885750496139, −8.280399796158393332053717499658, −8.239756642958642793733252900251, −7.59730763785553977297669688816, −6.69815938826210976497182460054, −6.42641935955634355910478692769, −6.26516763356391759809440579277, −5.59373849794804794283202668866, −5.24942327993844182806268054465, −4.07272256239174100808766529754, −3.75367125609998734303212785279, −3.33696487171546773939041879927, −2.58493787539929145083893494346, −1.23858427635752238422204537063, −0.70329762459793806379514394512,
0.70329762459793806379514394512, 1.23858427635752238422204537063, 2.58493787539929145083893494346, 3.33696487171546773939041879927, 3.75367125609998734303212785279, 4.07272256239174100808766529754, 5.24942327993844182806268054465, 5.59373849794804794283202668866, 6.26516763356391759809440579277, 6.42641935955634355910478692769, 6.69815938826210976497182460054, 7.59730763785553977297669688816, 8.239756642958642793733252900251, 8.280399796158393332053717499658, 9.391009028096546532885750496139, 9.580968173140962867665442901502, 9.722647143018227551827668992716, 10.41734914639437599321239868327, 10.81321270682472866927281974235, 11.21094356727767450999692234476
