| L(s) = 1 | − 16·4-s + 1.53e3·11-s + 256·16-s − 2.20e3·19-s − 1.12e4·29-s − 7.97e3·31-s − 3.08e3·41-s − 2.45e4·44-s + 3.31e4·49-s + 5.67e4·59-s + 1.10e4·61-s − 4.09e3·64-s − 8.47e4·71-s + 3.52e4·76-s + 7.92e4·79-s + 1.15e5·89-s + 2.82e5·101-s − 4.36e5·109-s + 1.79e5·116-s + 1.44e6·121-s + 1.27e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 3.82·11-s + 1/4·16-s − 1.39·19-s − 2.47·29-s − 1.49·31-s − 0.286·41-s − 1.91·44-s + 1.97·49-s + 2.12·59-s + 0.380·61-s − 1/8·64-s − 1.99·71-s + 0.699·76-s + 1.42·79-s + 1.54·89-s + 2.75·101-s − 3.52·109-s + 1.23·116-s + 8.98·121-s + 0.745·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.065449087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.065449087\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 33130 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 768 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 740470 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2696830 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1100 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8928490 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5610 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3988 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138667750 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1542 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 268756210 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 153278630 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 635715430 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 28380 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5522 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2088083650 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 42372 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1429023310 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 39640 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4298931010 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 57690 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 3671481410 T^{2} + p^{10} T^{4} \) |
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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
−10.45878641807936153144520275275, −10.01503247285661278905836971827, −9.286182846171536164162164129870, −9.115039257302729783277710030296, −8.988047704064253644969252529599, −8.513341555650354429176005815021, −7.69345642361947392981052280431, −7.22502952621022028078980257845, −6.62321053354619665675403130455, −6.56123098507550203549799057213, −5.65476732915159716365399131516, −5.59082569773693072920662293370, −4.38704126691805341040330892708, −4.21918009388164415945918036430, −3.62089652125065105625473922294, −3.54532930482973513039849491380, −2.05839398075179224184750172215, −1.83666492989126058309644772393, −1.08852729761872246073357427869, −0.47154443978581384830786165414,
0.47154443978581384830786165414, 1.08852729761872246073357427869, 1.83666492989126058309644772393, 2.05839398075179224184750172215, 3.54532930482973513039849491380, 3.62089652125065105625473922294, 4.21918009388164415945918036430, 4.38704126691805341040330892708, 5.59082569773693072920662293370, 5.65476732915159716365399131516, 6.56123098507550203549799057213, 6.62321053354619665675403130455, 7.22502952621022028078980257845, 7.69345642361947392981052280431, 8.513341555650354429176005815021, 8.988047704064253644969252529599, 9.115039257302729783277710030296, 9.286182846171536164162164129870, 10.01503247285661278905836971827, 10.45878641807936153144520275275
