| L(s) = 1 | + 5-s + 4·7-s − 4·13-s − 4·19-s + 6·23-s + 25-s + 6·29-s + 8·31-s + 4·35-s − 2·37-s + 6·41-s + 8·43-s − 6·47-s + 9·49-s − 6·53-s − 12·59-s + 2·61-s − 4·65-s + 10·67-s + 12·71-s + 16·73-s − 8·79-s − 6·89-s − 16·91-s − 4·95-s + 14·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.10·13-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s + 0.937·41-s + 1.21·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 1.22·67-s + 1.42·71-s + 1.87·73-s − 0.900·79-s − 0.635·89-s − 1.67·91-s − 0.410·95-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.609357620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.609357620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.55659688927925, −12.22034177509009, −11.40470503289055, −11.33091098159230, −10.77007105583990, −10.30835062599800, −9.916195040304457, −9.282356764997255, −8.951912274974255, −8.326736364710535, −8.024137606385770, −7.571206971591114, −7.030449482364371, −6.442185268673946, −6.156494375890461, −5.289241814871944, −5.003751026499569, −4.588674121189186, −4.274991599574984, −3.354178788714292, −2.752806442526217, −2.264000204916901, −1.844295927365790, −1.052643798591185, −0.6214821929040408,
0.6214821929040408, 1.052643798591185, 1.844295927365790, 2.264000204916901, 2.752806442526217, 3.354178788714292, 4.274991599574984, 4.588674121189186, 5.003751026499569, 5.289241814871944, 6.156494375890461, 6.442185268673946, 7.030449482364371, 7.571206971591114, 8.024137606385770, 8.326736364710535, 8.951912274974255, 9.282356764997255, 9.916195040304457, 10.30835062599800, 10.77007105583990, 11.33091098159230, 11.40470503289055, 12.22034177509009, 12.55659688927925
