| L(s) = 1 | − 2·5-s + 4·11-s + 3·13-s − 7·17-s + 5·19-s + 4·23-s + 5·25-s − 29-s + 6·31-s − 11·37-s + 9·41-s − 5·43-s + 6·47-s + 3·53-s − 8·55-s − 14·59-s − 6·61-s − 6·65-s + 26·67-s + 16·71-s + 7·73-s − 18·79-s − 83-s + 14·85-s − 15·89-s − 10·95-s − 17·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 0.832·13-s − 1.69·17-s + 1.14·19-s + 0.834·23-s + 25-s − 0.185·29-s + 1.07·31-s − 1.80·37-s + 1.40·41-s − 0.762·43-s + 0.875·47-s + 0.412·53-s − 1.07·55-s − 1.82·59-s − 0.768·61-s − 0.744·65-s + 3.17·67-s + 1.89·71-s + 0.819·73-s − 2.02·79-s − 0.109·83-s + 1.51·85-s − 1.58·89-s − 1.02·95-s − 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.247567507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247567507\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.425227464756000798854648875019, −8.044008778603206265035216160699, −7.72968583124454813501522639041, −7.13062751110142116872727820950, −6.91262350179843331823121520073, −6.59450139267222933618571231482, −6.54803649366268306762141582378, −5.75310812878798172400534994717, −5.53528593985476362032981996572, −5.04978739854270182573705953432, −4.60093741361838950831408494071, −4.22715439865998572172339636866, −4.02646712820557168961653096035, −3.34759211510826943030556262876, −3.34188629903753066842655143238, −2.62606682154743824723359755987, −2.20166741297343334647315815421, −1.41890205762233065849731444528, −1.12742308660158117373413333974, −0.45174014862334022527628124690,
0.45174014862334022527628124690, 1.12742308660158117373413333974, 1.41890205762233065849731444528, 2.20166741297343334647315815421, 2.62606682154743824723359755987, 3.34188629903753066842655143238, 3.34759211510826943030556262876, 4.02646712820557168961653096035, 4.22715439865998572172339636866, 4.60093741361838950831408494071, 5.04978739854270182573705953432, 5.53528593985476362032981996572, 5.75310812878798172400534994717, 6.54803649366268306762141582378, 6.59450139267222933618571231482, 6.91262350179843331823121520073, 7.13062751110142116872727820950, 7.72968583124454813501522639041, 8.044008778603206265035216160699, 8.425227464756000798854648875019
