| L(s) = 1 | − 4·5-s + 4·7-s + 2·11-s + 10·13-s + 6·17-s + 4·19-s + 6·23-s + 5·25-s − 12·29-s + 7·31-s − 16·35-s − 7·37-s − 4·41-s − 14·43-s − 2·47-s + 9·49-s + 6·53-s − 8·55-s + 6·59-s + 9·61-s − 40·65-s + 7·67-s + 16·71-s − 10·73-s + 8·77-s − 79-s − 28·83-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.51·7-s + 0.603·11-s + 2.77·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 25-s − 2.22·29-s + 1.25·31-s − 2.70·35-s − 1.15·37-s − 0.624·41-s − 2.13·43-s − 0.291·47-s + 9/7·49-s + 0.824·53-s − 1.07·55-s + 0.781·59-s + 1.15·61-s − 4.96·65-s + 0.855·67-s + 1.89·71-s − 1.17·73-s + 0.911·77-s − 0.112·79-s − 3.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.636946431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.636946431\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
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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
−9.891103050997054822796870221026, −9.102578133149313291686741161861, −8.560629388506287812779049031959, −8.298330237285907210775313761393, −8.290977677570387414625359999282, −7.87074777864712119392797451388, −7.20213524718571884979162864814, −7.03371033028207807433694901342, −6.63604349944254916072148666766, −5.76266706380442238274247075175, −5.55159897711243042775569830405, −5.17272820491445669458447354842, −4.57937439209619076441370522601, −3.95378218285554347177572472636, −3.60755289181998327364524526290, −3.59989910700719067871166672192, −2.88706280955806689260401758043, −1.56298052508001374066927580267, −1.47932143978915576995787333997, −0.74239155082555296255416921322,
0.74239155082555296255416921322, 1.47932143978915576995787333997, 1.56298052508001374066927580267, 2.88706280955806689260401758043, 3.59989910700719067871166672192, 3.60755289181998327364524526290, 3.95378218285554347177572472636, 4.57937439209619076441370522601, 5.17272820491445669458447354842, 5.55159897711243042775569830405, 5.76266706380442238274247075175, 6.63604349944254916072148666766, 7.03371033028207807433694901342, 7.20213524718571884979162864814, 7.87074777864712119392797451388, 8.290977677570387414625359999282, 8.298330237285907210775313761393, 8.560629388506287812779049031959, 9.102578133149313291686741161861, 9.891103050997054822796870221026
