| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 2·13-s + 16-s + 2·20-s − 25-s + 2·26-s + 4·31-s + 32-s − 6·37-s + 2·40-s − 2·41-s + 4·43-s + 6·49-s − 50-s + 2·52-s + 4·53-s + 4·62-s + 64-s + 4·65-s + 20·67-s + 8·71-s − 6·74-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.554·13-s + 1/4·16-s + 0.447·20-s − 1/5·25-s + 0.392·26-s + 0.718·31-s + 0.176·32-s − 0.986·37-s + 0.316·40-s − 0.312·41-s + 0.609·43-s + 6/7·49-s − 0.141·50-s + 0.277·52-s + 0.549·53-s + 0.508·62-s + 1/8·64-s + 0.496·65-s + 2.44·67-s + 0.949·71-s − 0.697·74-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.455314771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.455314771\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.992923436792907318404480712483, −8.267121893852668072322600252327, −8.163415009957602753855022303875, −7.35000497730763797828421881048, −6.88620236694935056345201968295, −6.46111627855734693606423617191, −6.00012058811656471583127714834, −5.44935345913756341642855777773, −5.17409208128022102155370988850, −4.43900568871848024480247765676, −3.83614887703557357904574380588, −3.36404350310135243902152774264, −2.48182597387038766900016225744, −2.04734299350582736873840896655, −1.07626451240917742437642025844,
1.07626451240917742437642025844, 2.04734299350582736873840896655, 2.48182597387038766900016225744, 3.36404350310135243902152774264, 3.83614887703557357904574380588, 4.43900568871848024480247765676, 5.17409208128022102155370988850, 5.44935345913756341642855777773, 6.00012058811656471583127714834, 6.46111627855734693606423617191, 6.88620236694935056345201968295, 7.35000497730763797828421881048, 8.163415009957602753855022303875, 8.267121893852668072322600252327, 8.992923436792907318404480712483
