L(s) = 1 | + 4·5-s − 9-s − 12·19-s + 11·25-s + 16·29-s − 16·31-s − 12·41-s − 4·45-s − 2·49-s + 24·59-s + 28·61-s − 16·71-s + 24·79-s + 81-s − 12·89-s − 48·95-s + 4·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s − 64·155-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 1/3·9-s − 2.75·19-s + 11/5·25-s + 2.97·29-s − 2.87·31-s − 1.87·41-s − 0.596·45-s − 2/7·49-s + 3.12·59-s + 3.58·61-s − 1.89·71-s + 2.70·79-s + 1/9·81-s − 1.27·89-s − 4.92·95-s + 0.383·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.741740558\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.741740558\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.406171627091146240017670032624, −8.852788276641653995601695500925, −8.677837639968309451837005472450, −8.392865466590526959443821282965, −8.121732002480809353675107529992, −7.21516605585194194959190067794, −6.79989865345280484695001224396, −6.62294726370191494405706318762, −6.37911386646073402901520764011, −5.63860816953378232698023246273, −5.57182273236609865220494082500, −4.95065225135551634082598349079, −4.73602281260995820595505859434, −3.83291737925194375357772197923, −3.76119844539937844423297141054, −2.79139402660676206332875153889, −2.39645388668815366288858194558, −2.05373433774555767093572023145, −1.52897513463252493945054421534, −0.58184425971407767233654672679,
0.58184425971407767233654672679, 1.52897513463252493945054421534, 2.05373433774555767093572023145, 2.39645388668815366288858194558, 2.79139402660676206332875153889, 3.76119844539937844423297141054, 3.83291737925194375357772197923, 4.73602281260995820595505859434, 4.95065225135551634082598349079, 5.57182273236609865220494082500, 5.63860816953378232698023246273, 6.37911386646073402901520764011, 6.62294726370191494405706318762, 6.79989865345280484695001224396, 7.21516605585194194959190067794, 8.121732002480809353675107529992, 8.392865466590526959443821282965, 8.677837639968309451837005472450, 8.852788276641653995601695500925, 9.406171627091146240017670032624