| L(s) = 1 | − 7·3-s + 5·5-s + 7·7-s + 27·9-s − 58·11-s + 164·13-s − 35·15-s − 50·17-s − 64·19-s − 49·21-s + 111·23-s − 224·27-s + 206·29-s + 130·31-s + 406·33-s + 35·35-s − 376·37-s − 1.14e3·39-s − 614·41-s − 394·43-s + 135·45-s − 120·47-s − 294·49-s + 350·51-s + 508·53-s − 290·55-s + 448·57-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s + 0.377·7-s + 9-s − 1.58·11-s + 3.49·13-s − 0.602·15-s − 0.713·17-s − 0.772·19-s − 0.509·21-s + 1.00·23-s − 1.59·27-s + 1.31·29-s + 0.753·31-s + 2.14·33-s + 0.169·35-s − 1.67·37-s − 4.71·39-s − 2.33·41-s − 1.39·43-s + 0.447·45-s − 0.372·47-s − 6/7·49-s + 0.960·51-s + 1.31·53-s − 0.710·55-s + 1.04·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.398215353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398215353\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 58 T + 2033 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 50 T - 2413 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 64 T - 2763 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 111 T + 154 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 103 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 130 T - 12891 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 376 T + 90723 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 307 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 197 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 120 T - 89423 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 508 T + 109187 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 600 T + 154621 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 165 T - 199756 T^{2} - 165 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 633 T + 99926 T^{2} - 633 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 606 T - 21781 T^{2} + 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1316 T + 1238817 T^{2} - 1316 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 61 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 187 T - 670000 T^{2} - 187 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 406 T + p^{3} T^{2} )^{2} \) |
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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
−11.59560752761809697768659222039, −10.99964615122909534875762087810, −10.89034930035001721152852817513, −10.36271321260575420694687390688, −10.18100299253983700917528019161, −9.248393436704561328086756911518, −8.612740752605213903175287420079, −8.284062543157981878091260833824, −8.080120498539775468298745083400, −6.76478147636842431553882527762, −6.73044293522881481754444403738, −6.15926143775471184016708559026, −5.67745049379818177983647645963, −4.95362632355586402340829423380, −4.89795716314073745381313288440, −3.69857540536067335720712030590, −3.36276871556921022259308770403, −2.08338995306755589223515431297, −1.41318428586734288907352820178, −0.52185977981391718091680818150,
0.52185977981391718091680818150, 1.41318428586734288907352820178, 2.08338995306755589223515431297, 3.36276871556921022259308770403, 3.69857540536067335720712030590, 4.89795716314073745381313288440, 4.95362632355586402340829423380, 5.67745049379818177983647645963, 6.15926143775471184016708559026, 6.73044293522881481754444403738, 6.76478147636842431553882527762, 8.080120498539775468298745083400, 8.284062543157981878091260833824, 8.612740752605213903175287420079, 9.248393436704561328086756911518, 10.18100299253983700917528019161, 10.36271321260575420694687390688, 10.89034930035001721152852817513, 10.99964615122909534875762087810, 11.59560752761809697768659222039
