| L(s) = 1 | − 2·2-s + 8·4-s + 5·5-s − 40·8-s − 10·10-s + 10·11-s + 80·13-s + 80·16-s − 14·17-s − 226·19-s + 40·20-s − 20·22-s − 81·23-s − 160·26-s − 220·29-s + 189·31-s − 320·32-s + 28·34-s + 340·37-s + 452·38-s − 200·40-s − 130·41-s − 10·43-s + 80·44-s + 162·46-s + 160·47-s + 343·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s + 0.447·5-s − 1.76·8-s − 0.316·10-s + 0.274·11-s + 1.70·13-s + 5/4·16-s − 0.199·17-s − 2.72·19-s + 0.447·20-s − 0.193·22-s − 0.734·23-s − 1.20·26-s − 1.40·29-s + 1.09·31-s − 1.76·32-s + 0.141·34-s + 1.51·37-s + 1.92·38-s − 0.790·40-s − 0.495·41-s − 0.0354·43-s + 0.274·44-s + 0.519·46-s + 0.496·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9598587748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9598587748\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T - p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T - 1231 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 80 T + 4203 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 113 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 81 T - 5606 T^{2} + 81 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 220 T + 24011 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 189 T + 5930 T^{2} - 189 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 170 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 130 T - 52021 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T - 79407 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 160 T - 78223 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 631 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 560 T + 108221 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 229 T - 174540 T^{2} + 229 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 750 T + 261737 T^{2} + 750 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 890 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 890 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 27 T - 492310 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 429 T - 387746 T^{2} - 429 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 750 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1480 T + 1277727 T^{2} - 1480 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.26008293152735235843512638596, −10.51499341841510425066252083858, −10.33444606673378017467034090029, −9.624566282480676071864457500965, −9.082491637462834796069432351414, −8.712417732327181926060636556110, −8.658392865422627913971931404620, −7.75404814360569312337159741284, −7.56009622848713520751142647571, −6.54641831002292432314467491172, −6.21628746759172109763121061673, −6.16509011174810456904155011425, −5.72247333173425993509518353084, −4.43318349469858688475660762357, −4.23599135948179247971190531467, −3.26557146174874510791562678759, −2.82606889653381361140178983143, −1.88484135250783116388313116506, −1.61419533903029841439709484280, −0.34921924540581454492748143130,
0.34921924540581454492748143130, 1.61419533903029841439709484280, 1.88484135250783116388313116506, 2.82606889653381361140178983143, 3.26557146174874510791562678759, 4.23599135948179247971190531467, 4.43318349469858688475660762357, 5.72247333173425993509518353084, 6.16509011174810456904155011425, 6.21628746759172109763121061673, 6.54641831002292432314467491172, 7.56009622848713520751142647571, 7.75404814360569312337159741284, 8.658392865422627913971931404620, 8.712417732327181926060636556110, 9.082491637462834796069432351414, 9.624566282480676071864457500965, 10.33444606673378017467034090029, 10.51499341841510425066252083858, 11.26008293152735235843512638596
