| L(s) = 1 | − 3·3-s − 4·5-s + 6·9-s + 4·13-s + 12·15-s + 11·25-s − 9·27-s + 4·31-s − 12·37-s − 12·39-s − 7·41-s + 2·43-s − 24·45-s + 9·49-s + 16·53-s − 16·65-s + 19·67-s + 4·71-s − 33·75-s + 20·79-s + 9·81-s − 6·83-s − 16·89-s − 12·93-s + 17·107-s + 36·111-s + 24·117-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s + 2·9-s + 1.10·13-s + 3.09·15-s + 11/5·25-s − 1.73·27-s + 0.718·31-s − 1.97·37-s − 1.92·39-s − 1.09·41-s + 0.304·43-s − 3.57·45-s + 9/7·49-s + 2.19·53-s − 1.98·65-s + 2.32·67-s + 0.474·71-s − 3.81·75-s + 2.25·79-s + 81-s − 0.658·83-s − 1.69·89-s − 1.24·93-s + 1.64·107-s + 3.41·111-s + 2.21·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5919604831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5919604831\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 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
−8.395554371954880583317378328458, −8.141423178222912002946182005187, −7.30500629447988124749823339787, −7.13209011983909208434578048260, −6.70911014125182219050222409484, −6.28370317004655490254917114244, −5.56793529064462917598221137428, −5.31048168972593050462843013818, −4.71611183763722531252205441084, −4.24901043381761722054389757838, −3.63897286269102582264307587050, −3.49733338088313715165637828644, −2.30303070616410154178368382639, −1.17683136507181610287054503120, −0.53379195357559506106793994146,
0.53379195357559506106793994146, 1.17683136507181610287054503120, 2.30303070616410154178368382639, 3.49733338088313715165637828644, 3.63897286269102582264307587050, 4.24901043381761722054389757838, 4.71611183763722531252205441084, 5.31048168972593050462843013818, 5.56793529064462917598221137428, 6.28370317004655490254917114244, 6.70911014125182219050222409484, 7.13209011983909208434578048260, 7.30500629447988124749823339787, 8.141423178222912002946182005187, 8.395554371954880583317378328458
