| L(s) = 1 | + 8·7-s + 36·19-s − 20·25-s − 136·43-s − 156·49-s + 344·61-s − 408·73-s − 164·121-s + 127-s + 131-s + 288·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 116·169-s + 173-s − 160·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 1.89·19-s − 4/5·25-s − 3.16·43-s − 3.18·49-s + 5.63·61-s − 5.58·73-s − 1.35·121-s + 0.00787·127-s + 0.00763·131-s + 2.16·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.686·169-s + 0.00578·173-s − 0.914·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.482792104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.482792104\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 p T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1018 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2}( 1 + 54 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1642 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 218 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2914 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 2342 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1586 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6514 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 86 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 4498 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 6046 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 102 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 1238 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13138 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 286 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17698 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.04206516715673124976368407260, −5.91166197553396410834968300021, −5.64120045955736703901843883743, −5.25041553340672697018310189135, −5.17732237737954275526170252390, −5.16946338933464751890969143085, −4.88595796915055881368266629693, −4.79166079622327055801876374749, −4.36175489445693716198396646950, −4.24923201933247850107893525905, −3.95310631873343617423378791985, −3.65403297737527322059683536036, −3.59875281124650053037433071774, −3.26632079849925321636278332623, −3.08558685329103284496952278839, −2.82933440500997712516385050159, −2.61125448105746996507572062061, −2.15235594742540907913175186086, −2.10071631984564771185860853270, −1.62673324984187024108521267146, −1.42483193429736491924522441120, −1.31213221796337293787367660065, −1.09025906377026233034097989215, −0.35222193322202729987639622234, −0.30540138204274482205725811347,
0.30540138204274482205725811347, 0.35222193322202729987639622234, 1.09025906377026233034097989215, 1.31213221796337293787367660065, 1.42483193429736491924522441120, 1.62673324984187024108521267146, 2.10071631984564771185860853270, 2.15235594742540907913175186086, 2.61125448105746996507572062061, 2.82933440500997712516385050159, 3.08558685329103284496952278839, 3.26632079849925321636278332623, 3.59875281124650053037433071774, 3.65403297737527322059683536036, 3.95310631873343617423378791985, 4.24923201933247850107893525905, 4.36175489445693716198396646950, 4.79166079622327055801876374749, 4.88595796915055881368266629693, 5.16946338933464751890969143085, 5.17732237737954275526170252390, 5.25041553340672697018310189135, 5.64120045955736703901843883743, 5.91166197553396410834968300021, 6.04206516715673124976368407260
