| L(s) = 1 | + 4·5-s − 3·7-s + 28·11-s + 11·13-s − 88·17-s + 58·19-s + 172·23-s + 125·25-s + 192·29-s − 116·31-s − 12·35-s − 138·37-s + 384·41-s − 328·43-s + 156·47-s + 343·49-s + 784·53-s + 112·55-s + 412·59-s + 425·61-s + 44·65-s − 257·67-s + 2.00e3·71-s − 718·73-s − 84·77-s − 877·79-s − 328·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 0.161·7-s + 0.767·11-s + 0.234·13-s − 1.25·17-s + 0.700·19-s + 1.55·23-s + 25-s + 1.22·29-s − 0.672·31-s − 0.0579·35-s − 0.613·37-s + 1.46·41-s − 1.16·43-s + 0.484·47-s + 49-s + 2.03·53-s + 0.274·55-s + 0.909·59-s + 0.892·61-s + 0.0839·65-s − 0.468·67-s + 3.34·71-s − 1.15·73-s − 0.124·77-s − 1.24·79-s − 0.433·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.333721354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.333721354\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T - 109 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T - 334 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 28 T - 547 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 11 T - 2076 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 44 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 172 T + 17417 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 192 T + 12475 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 116 T - 16335 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 69 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 384 T + 78535 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 328 T + 28077 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 156 T - 79487 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 392 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 412 T - 35635 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 425 T - 46356 T^{2} - 425 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 257 T - 234714 T^{2} + 257 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1000 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 359 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 877 T + 276090 T^{2} + 877 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 328 T - 464203 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1572 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1483 T + 1286616 T^{2} - 1483 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.22506019527761033686665983662, −10.14216929907730292376989992801, −9.350130940176037922680062527259, −9.020534435284298184562811431769, −8.678281800984904516333734871005, −8.544202177901722602809517226757, −7.54691566322269069005738199093, −7.24650867154556919791910586934, −6.73589275710730070435016942947, −6.54511044920518010015174810621, −5.82734509053004913085101402410, −5.43436024597814758713018789838, −4.63965408864599461481208023207, −4.62963590334389746907152016623, −3.50088807492538801063425082066, −3.46768152953684714389152572140, −2.41652323201728577711516370380, −2.13974071610175097490448164495, −0.933253730624204428339741375135, −0.804159124062839267316897135724,
0.804159124062839267316897135724, 0.933253730624204428339741375135, 2.13974071610175097490448164495, 2.41652323201728577711516370380, 3.46768152953684714389152572140, 3.50088807492538801063425082066, 4.62963590334389746907152016623, 4.63965408864599461481208023207, 5.43436024597814758713018789838, 5.82734509053004913085101402410, 6.54511044920518010015174810621, 6.73589275710730070435016942947, 7.24650867154556919791910586934, 7.54691566322269069005738199093, 8.544202177901722602809517226757, 8.678281800984904516333734871005, 9.020534435284298184562811431769, 9.350130940176037922680062527259, 10.14216929907730292376989992801, 10.22506019527761033686665983662
