L(s) = 1 | − 2·5-s − 9-s − 10·11-s − 8·19-s − 25-s + 8·29-s + 22·41-s + 2·45-s − 11·49-s + 20·55-s + 24·59-s − 14·61-s + 14·71-s − 10·79-s + 81-s + 6·89-s + 16·95-s + 10·99-s − 12·101-s + 24·109-s + 53·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 1/3·9-s − 3.01·11-s − 1.83·19-s − 1/5·25-s + 1.48·29-s + 3.43·41-s + 0.298·45-s − 1.57·49-s + 2.69·55-s + 3.12·59-s − 1.79·61-s + 1.66·71-s − 1.12·79-s + 1/9·81-s + 0.635·89-s + 1.64·95-s + 1.00·99-s − 1.19·101-s + 2.29·109-s + 4.81·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4882381865\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4882381865\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−8.591047949550075871478787133855, −8.441201472848371988063470334681, −8.110090927072728382997584438213, −7.84380988510469127236053456005, −7.47621373781960526703507241229, −7.22991032648858796782306494284, −6.56008882336602238494155913746, −6.22408257724713854108591686867, −5.84245627309651084249190707967, −5.42663014605845049925119284201, −4.93516859292240993268529298400, −4.71513143668610859049671462006, −4.18546769875175989756241314618, −3.88979970121191682628409773550, −3.15540817771894792163069548177, −2.77828962518695630275832381903, −2.35301006351708017803347011602, −2.14176505761691054364407761983, −0.935039507705742142073608445260, −0.25789716590951574057422994658,
0.25789716590951574057422994658, 0.935039507705742142073608445260, 2.14176505761691054364407761983, 2.35301006351708017803347011602, 2.77828962518695630275832381903, 3.15540817771894792163069548177, 3.88979970121191682628409773550, 4.18546769875175989756241314618, 4.71513143668610859049671462006, 4.93516859292240993268529298400, 5.42663014605845049925119284201, 5.84245627309651084249190707967, 6.22408257724713854108591686867, 6.56008882336602238494155913746, 7.22991032648858796782306494284, 7.47621373781960526703507241229, 7.84380988510469127236053456005, 8.110090927072728382997584438213, 8.441201472848371988063470334681, 8.591047949550075871478787133855