L(s) = 1 | + 5-s + 7-s + 6·11-s + 13-s − 2·19-s + 6·23-s + 6·29-s − 8·31-s + 35-s − 14·37-s − 6·41-s + 4·43-s + 12·47-s + 7·49-s + 12·53-s + 6·55-s − 11·61-s + 65-s + 7·67-s + 12·71-s + 22·73-s + 6·77-s + 79-s + 6·83-s + 24·89-s + 91-s − 2·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.80·11-s + 0.277·13-s − 0.458·19-s + 1.25·23-s + 1.11·29-s − 1.43·31-s + 0.169·35-s − 2.30·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 49-s + 1.64·53-s + 0.809·55-s − 1.40·61-s + 0.124·65-s + 0.855·67-s + 1.42·71-s + 2.57·73-s + 0.683·77-s + 0.112·79-s + 0.658·83-s + 2.54·89-s + 0.104·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.478193664\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.478193664\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.275862770202948730539458817071, −9.117103165356682067205774154988, −8.896083244660351379071625984535, −8.707507576078659200805731718439, −7.948433025806952808223705155257, −7.64949770793006575000847689344, −6.93452850895556673355372459868, −6.88927875059497253503524900796, −6.43878483426655247084553947299, −6.03255574087117669283223451061, −5.39609020589636824390682936124, −5.13815789471283498099003027906, −4.68973545747129347315240438732, −4.01050746590354684390878290993, −3.56683067622703208294517785351, −3.46317722579935583968584931004, −2.28209736958848734266206952127, −2.16232661081501294189568825430, −1.29058377336987598391147609536, −0.817107985060873369225355709621,
0.817107985060873369225355709621, 1.29058377336987598391147609536, 2.16232661081501294189568825430, 2.28209736958848734266206952127, 3.46317722579935583968584931004, 3.56683067622703208294517785351, 4.01050746590354684390878290993, 4.68973545747129347315240438732, 5.13815789471283498099003027906, 5.39609020589636824390682936124, 6.03255574087117669283223451061, 6.43878483426655247084553947299, 6.88927875059497253503524900796, 6.93452850895556673355372459868, 7.64949770793006575000847689344, 7.948433025806952808223705155257, 8.707507576078659200805731718439, 8.896083244660351379071625984535, 9.117103165356682067205774154988, 9.275862770202948730539458817071