| L(s) = 1 | + 5·4-s − 8·5-s − 20·7-s − 20·11-s + 9·16-s − 20·17-s + 38·19-s − 40·20-s + 40·23-s − 2·25-s − 100·28-s + 160·35-s − 20·43-s − 100·44-s + 160·47-s + 202·49-s + 160·55-s − 20·61-s − 35·64-s − 100·68-s − 20·73-s + 190·76-s + 400·77-s − 72·80-s − 140·83-s + 160·85-s + 200·92-s + ⋯ |
| L(s) = 1 | + 5/4·4-s − 8/5·5-s − 2.85·7-s − 1.81·11-s + 9/16·16-s − 1.17·17-s + 2·19-s − 2·20-s + 1.73·23-s − 0.0799·25-s − 3.57·28-s + 32/7·35-s − 0.465·43-s − 2.27·44-s + 3.40·47-s + 4.12·49-s + 2.90·55-s − 0.327·61-s − 0.546·64-s − 1.47·68-s − 0.273·73-s + 5/2·76-s + 5.19·77-s − 0.899·80-s − 1.68·83-s + 1.88·85-s + 2.17·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5706795528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5706795528\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 250 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 482 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1622 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2162 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3890 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5762 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 718 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12182 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 5042 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13010 T^{2} + p^{4} T^{4} \) |
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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
−13.18005902823510551846134302762, −12.18401821572814709674650577179, −11.89509263107345803239523861776, −11.26526740643542340514363645739, −10.86847771875901352206945949339, −10.37063547741588281306966559113, −9.846833566453739509846033159449, −9.269550919057457082656945567724, −8.812784056032985216387832331513, −7.78750784074357657709857095912, −7.53411458814331493580239582695, −6.87314669345715095981511118978, −6.85965820895496854444838613398, −5.84381933889310680248704828115, −5.43439370014349689230260483911, −4.27466738455366368628853552074, −3.51154564522615496703632905059, −2.88058184753101303400200901047, −2.69139019860051053460730107352, −0.42970539854350394378232943799,
0.42970539854350394378232943799, 2.69139019860051053460730107352, 2.88058184753101303400200901047, 3.51154564522615496703632905059, 4.27466738455366368628853552074, 5.43439370014349689230260483911, 5.84381933889310680248704828115, 6.85965820895496854444838613398, 6.87314669345715095981511118978, 7.53411458814331493580239582695, 7.78750784074357657709857095912, 8.812784056032985216387832331513, 9.269550919057457082656945567724, 9.846833566453739509846033159449, 10.37063547741588281306966559113, 10.86847771875901352206945949339, 11.26526740643542340514363645739, 11.89509263107345803239523861776, 12.18401821572814709674650577179, 13.18005902823510551846134302762
