| L(s) = 1 | − 4-s + 2·5-s − 9-s − 2·11-s + 16-s − 6·19-s − 2·20-s − 25-s − 12·29-s − 2·31-s + 36-s + 4·41-s + 2·44-s − 2·45-s − 11·49-s − 4·55-s + 28·59-s + 28·61-s − 64-s + 18·71-s + 6·76-s − 30·79-s + 2·80-s + 81-s + 2·89-s − 12·95-s + 2·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s − 1/3·9-s − 0.603·11-s + 1/4·16-s − 1.37·19-s − 0.447·20-s − 1/5·25-s − 2.22·29-s − 0.359·31-s + 1/6·36-s + 0.624·41-s + 0.301·44-s − 0.298·45-s − 1.57·49-s − 0.539·55-s + 3.64·59-s + 3.58·61-s − 1/8·64-s + 2.13·71-s + 0.688·76-s − 3.37·79-s + 0.223·80-s + 1/9·81-s + 0.211·89-s − 1.23·95-s + 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.316010079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316010079\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.28333496243343160019385547601, −9.811493788425270931391302038918, −9.533836145126046391854071597588, −9.027610487362500617128024115490, −8.671093947431050969266309820508, −8.106490942136047456984712173931, −8.040791268105960274438525677327, −7.24458161526450368608660492853, −6.81544539371626149443485786711, −6.49069888343439444986924777984, −5.65985739249986372350084822645, −5.55733500803930196949177533093, −5.27927439793987147849687201147, −4.50740365625074095957041183920, −3.82969445389007630682392656451, −3.73543790169156736936307645155, −2.69758095415948915636767956929, −2.21241908536177274458602143941, −1.76117529757177652857111550953, −0.52648655962142875337134174249,
0.52648655962142875337134174249, 1.76117529757177652857111550953, 2.21241908536177274458602143941, 2.69758095415948915636767956929, 3.73543790169156736936307645155, 3.82969445389007630682392656451, 4.50740365625074095957041183920, 5.27927439793987147849687201147, 5.55733500803930196949177533093, 5.65985739249986372350084822645, 6.49069888343439444986924777984, 6.81544539371626149443485786711, 7.24458161526450368608660492853, 8.040791268105960274438525677327, 8.106490942136047456984712173931, 8.671093947431050969266309820508, 9.027610487362500617128024115490, 9.533836145126046391854071597588, 9.811493788425270931391302038918, 10.28333496243343160019385547601
