| L(s) = 1 | − 180·17-s + 250·25-s − 1.04e3·41-s − 686·49-s − 860·73-s − 2.05e3·89-s − 3.82e3·97-s + 540·113-s + 2.33e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.56·17-s + 2·25-s − 3.97·41-s − 2·49-s − 1.37·73-s − 2.44·89-s − 3.99·97-s + 0.449·113-s + 1.75·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4792918372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4792918372\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2338 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2482 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 522 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 74914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 304958 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 596626 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 430 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 678926 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1026 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1910 T + p^{3} T^{2} )^{2} \) |
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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
−10.67016050514561331154485571560, −9.990546827521816250209878672378, −9.819188016137967151442135240238, −9.050586010279375456661197467882, −8.783472809410815713234616394584, −8.375186935866125219423757538266, −8.147892752543172415150220471379, −7.08345020592723827539044656372, −6.97289569395717397899705866122, −6.59067541847462047433441894719, −6.16734807862017247304645257915, −5.17684174673876907358549479866, −5.08308260030683094613814473702, −4.42637463352687941659332614256, −4.00340861063318723668598869248, −3.06510397444371321587764166741, −2.84269744984927898170892811505, −1.86993591028199744922661392953, −1.44723192460365610634002241786, −0.20003468293914237381110268328,
0.20003468293914237381110268328, 1.44723192460365610634002241786, 1.86993591028199744922661392953, 2.84269744984927898170892811505, 3.06510397444371321587764166741, 4.00340861063318723668598869248, 4.42637463352687941659332614256, 5.08308260030683094613814473702, 5.17684174673876907358549479866, 6.16734807862017247304645257915, 6.59067541847462047433441894719, 6.97289569395717397899705866122, 7.08345020592723827539044656372, 8.147892752543172415150220471379, 8.375186935866125219423757538266, 8.783472809410815713234616394584, 9.050586010279375456661197467882, 9.819188016137967151442135240238, 9.990546827521816250209878672378, 10.67016050514561331154485571560
