| L(s) = 1 | + 4.49e4·3-s − 1.04e6·4-s + 8.55e8·9-s − 4.70e10·12-s + 8.24e11·16-s + 3.60e13·25-s − 1.37e13·27-s − 3.21e14·31-s − 8.96e14·36-s − 2.51e15·37-s + 3.70e16·48-s + 2.27e16·49-s − 5.76e17·64-s − 8.80e17·67-s + 1.61e18·75-s − 1.61e18·81-s − 1.44e19·93-s − 2.40e19·97-s − 3.78e19·100-s − 3.10e19·103-s + 1.44e19·108-s − 1.12e20·111-s − 6.11e19·121-s + 3.36e20·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.31·3-s − 2·4-s + 0.735·9-s − 2.63·12-s + 3·16-s + 1.89·25-s − 0.347·27-s − 2.18·31-s − 1.47·36-s − 3.17·37-s + 3.95·48-s + 2·49-s − 4·64-s − 3.95·67-s + 2.49·75-s − 1.19·81-s − 2.87·93-s − 3.21·97-s − 3.78·100-s − 2.34·103-s + 0.695·108-s − 4.18·111-s − 121-s + 4.36·124-s + 2.20·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.1016207429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1016207429\) |
| \(L(\frac{21}{2})\) |
|
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 | $C_2$ | \( 1 - 44917 T + p^{19} T^{2} \) |
| 11 | $C_2$ | \( 1 + p^{19} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 8614293 T + p^{19} T^{2} )( 1 + 8614293 T + p^{19} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 15737834185533 T + p^{19} T^{2} )( 1 + 15737834185533 T + p^{19} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 160664629612495 T + p^{19} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1255665816556711 T + p^{19} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 15124546048050204 T + p^{19} T^{2} )( 1 + 15124546048050204 T + p^{19} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 46928500334852898 T + p^{19} T^{2} )( 1 + 46928500334852898 T + p^{19} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 72693638336311365 T + p^{19} T^{2} )( 1 + 72693638336311365 T + p^{19} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 440100989536856089 T + p^{19} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 197781989052854343 T + p^{19} T^{2} )( 1 + 197781989052854343 T + p^{19} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 150224765047129881 T + p^{19} T^{2} )( 1 + 150224765047129881 T + p^{19} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 12044200680599180239 T + p^{19} T^{2} )^{2} \) |
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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
−13.47764600864624705169176622649, −12.39581691858879317378584834619, −12.18858115053358865103965473144, −10.57961473800396617430806650382, −10.56512025217872590530884993560, −9.507484194078767770258516346278, −9.061816745923506316931524366256, −8.759655319405613987000677711360, −8.354127000547417613686525918433, −7.49221975842607888729890476327, −6.99273330642717595974915391185, −5.58473576638560199344546295307, −5.31820315234939264663792776087, −4.44575790069876911036563762872, −3.90039303383568225211976303943, −3.33640617977736921376000797751, −2.81309248432071230927723074986, −1.67432337328300400293285071008, −1.19481799607694825914755789357, −0.07511109273475328784116023800,
0.07511109273475328784116023800, 1.19481799607694825914755789357, 1.67432337328300400293285071008, 2.81309248432071230927723074986, 3.33640617977736921376000797751, 3.90039303383568225211976303943, 4.44575790069876911036563762872, 5.31820315234939264663792776087, 5.58473576638560199344546295307, 6.99273330642717595974915391185, 7.49221975842607888729890476327, 8.354127000547417613686525918433, 8.759655319405613987000677711360, 9.061816745923506316931524366256, 9.507484194078767770258516346278, 10.56512025217872590530884993560, 10.57961473800396617430806650382, 12.18858115053358865103965473144, 12.39581691858879317378584834619, 13.47764600864624705169176622649
