| L(s) = 1 | + 4-s − 2·13-s − 3·16-s + 12·17-s + 10·25-s − 12·29-s − 8·43-s + 2·49-s − 2·52-s − 12·53-s − 4·61-s − 7·64-s + 12·68-s − 16·79-s + 10·100-s − 12·101-s + 16·103-s − 24·107-s + 12·113-s − 12·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.554·13-s − 3/4·16-s + 2.91·17-s + 2·25-s − 2.22·29-s − 1.21·43-s + 2/7·49-s − 0.277·52-s − 1.64·53-s − 0.512·61-s − 7/8·64-s + 1.45·68-s − 1.80·79-s + 100-s − 1.19·101-s + 1.57·103-s − 2.32·107-s + 1.12·113-s − 1.11·116-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192130317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192130317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−14.25796646071668065415399511013, −12.94653876664632992175973565717, −12.92870767676208291067046516078, −12.12513433543839575821358493524, −11.87358826855113987605804394838, −11.16719593149364785823632598205, −10.74876269550658365604509198493, −10.10812231428556152192195977023, −9.615125411710301297732032956462, −9.132630457026905491932237417534, −8.340846955215312984274751734418, −7.71327862265261631414445850877, −7.26576752323852867591516307774, −6.72093424702019735750393022276, −5.82159304803056284359610985675, −5.34017448720661284081589497718, −4.60565980587590345138887976369, −3.47018727620373522194747060402, −2.91134212354398266583078098641, −1.57686584716747167281817741029,
1.57686584716747167281817741029, 2.91134212354398266583078098641, 3.47018727620373522194747060402, 4.60565980587590345138887976369, 5.34017448720661284081589497718, 5.82159304803056284359610985675, 6.72093424702019735750393022276, 7.26576752323852867591516307774, 7.71327862265261631414445850877, 8.340846955215312984274751734418, 9.132630457026905491932237417534, 9.615125411710301297732032956462, 10.10812231428556152192195977023, 10.74876269550658365604509198493, 11.16719593149364785823632598205, 11.87358826855113987605804394838, 12.12513433543839575821358493524, 12.92870767676208291067046516078, 12.94653876664632992175973565717, 14.25796646071668065415399511013
