| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 2·11-s + 15-s − 2·17-s + 2·21-s − 8·23-s + 25-s − 27-s − 2·33-s + 2·35-s + 4·37-s + 8·41-s + 6·43-s − 45-s − 8·47-s − 3·49-s + 2·51-s − 10·53-s − 2·55-s − 8·59-s − 2·61-s − 2·63-s + 8·69-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.258·15-s − 0.485·17-s + 0.436·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.348·33-s + 0.338·35-s + 0.657·37-s + 1.24·41-s + 0.914·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.280·51-s − 1.37·53-s − 0.269·55-s − 1.04·59-s − 0.256·61-s − 0.251·63-s + 0.963·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3469268437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3469268437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.45977793783298, −12.19379299696375, −11.62084594829255, −11.23549275730637, −10.84083743768355, −10.33480723150869, −9.774899107414703, −9.395649616557565, −9.120401433693890, −8.319675015894902, −7.917227482294149, −7.538241371417994, −6.865697583072545, −6.472211173002201, −6.031409698469904, −5.780020244770819, −4.908599873114431, −4.454307724081260, −4.078164510075542, −3.518308233738156, −2.982644229903113, −2.314166151976587, −1.663519367388139, −1.015041681683283, −0.1780199181726390,
0.1780199181726390, 1.015041681683283, 1.663519367388139, 2.314166151976587, 2.982644229903113, 3.518308233738156, 4.078164510075542, 4.454307724081260, 4.908599873114431, 5.780020244770819, 6.031409698469904, 6.472211173002201, 6.865697583072545, 7.538241371417994, 7.917227482294149, 8.319675015894902, 9.120401433693890, 9.395649616557565, 9.774899107414703, 10.33480723150869, 10.84083743768355, 11.23549275730637, 11.62084594829255, 12.19379299696375, 12.45977793783298
