| L(s) = 1 | + 5.90e4·3-s − 5.24e5·4-s + 9.68e7·7-s + 2.32e9·9-s − 3.09e10·12-s + 2.04e12·19-s + 5.71e12·21-s + 1.90e13·25-s + 6.86e13·27-s − 5.07e13·28-s − 4.79e14·31-s − 1.21e15·36-s + 1.58e15·37-s + 1.20e16·43-s − 2.01e15·49-s + 1.20e17·57-s + 2.02e17·61-s + 2.25e17·63-s + 1.44e17·64-s + 4.41e17·67-s − 2.86e17·73-s + 1.12e18·75-s − 1.07e18·76-s − 2.62e17·79-s + 1.35e18·81-s − 2.99e18·84-s − 2.82e19·93-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 0.907·7-s + 2·9-s − 1.73·12-s + 1.45·19-s + 1.57·21-s + 25-s + 1.73·27-s − 0.907·28-s − 3.25·31-s − 2·36-s + 1.99·37-s + 3.64·43-s − 0.177·49-s + 2.52·57-s + 2.21·61-s + 1.81·63-s + 64-s + 1.98·67-s − 0.569·73-s + 1.73·75-s − 1.45·76-s − 0.246·79-s + 81-s − 1.57·84-s − 5.63·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(6.277179504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.277179504\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p^{10} T + p^{19} T^{2} \) |
| 7 | $C_2$ | \( 1 - 96856453 T + p^{19} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p^{19} T^{2} + p^{38} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{19} T^{2} + p^{38} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{19} T^{2} + p^{38} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7913874521 T + p^{19} T^{2} )( 1 + 7913874521 T + p^{19} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p^{19} T^{2} + p^{38} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2300181624971 T + p^{19} T^{2} )( 1 + 252454137272 T + p^{19} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{19} T^{2} + p^{38} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 189181140858356 T + p^{19} T^{2} )( 1 + 289961649705409 T + p^{19} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 800421296193277 T + p^{19} T^{2} )( 1 - 780590048220370 T + p^{19} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6012529514786335 T + p^{19} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{19} T^{2} + p^{38} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{19} T^{2} + p^{38} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{19} T^{2} + p^{38} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 171366248956165199 T + p^{19} T^{2} )( 1 - 30901601965099333 T + p^{19} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 273531389481429392 T + p^{19} T^{2} )( 1 - 167643129887418785 T + p^{19} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 352850417428850191 T + p^{19} T^{2} )( 1 + 639446467598381530 T + p^{19} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 1699851029529528053 T + p^{19} T^{2} )( 1 + 1962240154233379276 T + p^{19} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{19} T^{2} + p^{38} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9502640950925166898 T + p^{19} T^{2} )( 1 + 9502640950925166898 T + p^{19} T^{2} ) \) |
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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
−14.21588995834844244517892802251, −13.64612183099239223210685742333, −12.78414203779864028188276129199, −12.73358025340843565084821453713, −11.31751567418832083638585903401, −10.89251528305971152572764262783, −9.642439526416713733468941961207, −9.400428685343997025739366777072, −8.861300666120112080790741056391, −8.201421185858428547167459474895, −7.54060035481372786026296431734, −7.13682848223445912008143925225, −5.63023582551122440147807062809, −4.99621583612341112379524321898, −4.07941032490452674832954258172, −3.78894834793450174241616401295, −2.75231405181806532155629499226, −2.19353664885276320573602748104, −1.26911196060362356602565407853, −0.67109127058146114767755007230,
0.67109127058146114767755007230, 1.26911196060362356602565407853, 2.19353664885276320573602748104, 2.75231405181806532155629499226, 3.78894834793450174241616401295, 4.07941032490452674832954258172, 4.99621583612341112379524321898, 5.63023582551122440147807062809, 7.13682848223445912008143925225, 7.54060035481372786026296431734, 8.201421185858428547167459474895, 8.861300666120112080790741056391, 9.400428685343997025739366777072, 9.642439526416713733468941961207, 10.89251528305971152572764262783, 11.31751567418832083638585903401, 12.73358025340843565084821453713, 12.78414203779864028188276129199, 13.64612183099239223210685742333, 14.21588995834844244517892802251
