| L(s) = 1 | + 5·7-s − 5·9-s + 9·23-s + 8·25-s + 8·31-s − 12·41-s − 6·47-s + 7·49-s − 25·63-s − 18·71-s + 10·73-s + 11·79-s + 16·81-s − 3·89-s + 19·97-s + 12·103-s − 3·113-s − 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 45·161-s + 163-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 5/3·9-s + 1.87·23-s + 8/5·25-s + 1.43·31-s − 1.87·41-s − 0.875·47-s + 49-s − 3.14·63-s − 2.13·71-s + 1.17·73-s + 1.23·79-s + 16/9·81-s − 0.317·89-s + 1.92·97-s + 1.18·103-s − 0.282·113-s − 0.636·121-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 + 0.0798·157-s + 3.54·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.761823671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761823671\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 103 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 13 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−9.242412338214102015719350391405, −8.951525192539982448844251810580, −8.523900641817344502700820740063, −8.099434635134167665772827084980, −7.83359059880360854260524300929, −6.91723516085483281440880391971, −6.64376725072407482350419078455, −5.85921338008075421167629211285, −5.19517632194054792159578724834, −4.89666732031064393153315678930, −4.56487927691487028929067771424, −3.28169182930450414726576490605, −2.97347628967119628865616056857, −2.03018378358717343211115514335, −1.04494068506100383113500362781,
1.04494068506100383113500362781, 2.03018378358717343211115514335, 2.97347628967119628865616056857, 3.28169182930450414726576490605, 4.56487927691487028929067771424, 4.89666732031064393153315678930, 5.19517632194054792159578724834, 5.85921338008075421167629211285, 6.64376725072407482350419078455, 6.91723516085483281440880391971, 7.83359059880360854260524300929, 8.099434635134167665772827084980, 8.523900641817344502700820740063, 8.951525192539982448844251810580, 9.242412338214102015719350391405
