| L(s) = 1 | + 2·5-s − 7-s − 3·9-s − 4·11-s − 4·13-s − 8·17-s − 2·19-s − 23-s − 25-s − 2·29-s + 6·31-s − 2·35-s + 10·37-s + 6·41-s − 8·43-s − 6·45-s − 6·47-s + 49-s − 2·53-s − 8·55-s − 10·61-s + 3·63-s − 8·65-s + 8·67-s + 12·71-s + 6·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s − 1.10·13-s − 1.94·17-s − 0.458·19-s − 0.208·23-s − 1/5·25-s − 0.371·29-s + 1.07·31-s − 0.338·35-s + 1.64·37-s + 0.937·41-s − 1.21·43-s − 0.894·45-s − 0.875·47-s + 1/7·49-s − 0.274·53-s − 1.07·55-s − 1.28·61-s + 0.377·63-s − 0.992·65-s + 0.977·67-s + 1.42·71-s + 0.702·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8595069548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8595069548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.81232642868411, −15.97191325793299, −15.43871991917556, −14.89933853811091, −14.27730521318330, −13.61037165421069, −13.18447493468213, −12.79332542766678, −11.93256831313707, −11.29519439150065, −10.74214754469468, −10.12527629531165, −9.464934763326976, −9.106130057765611, −8.118816119322836, −7.847938995231067, −6.682018738091914, −6.358313714918396, −5.611884469297738, −4.945122243136318, −4.359265353190208, −3.142395995439173, −2.366536424245905, −2.159249164233654, −0.3959817354564118,
0.3959817354564118, 2.159249164233654, 2.366536424245905, 3.142395995439173, 4.359265353190208, 4.945122243136318, 5.611884469297738, 6.358313714918396, 6.682018738091914, 7.847938995231067, 8.118816119322836, 9.106130057765611, 9.464934763326976, 10.12527629531165, 10.74214754469468, 11.29519439150065, 11.93256831313707, 12.79332542766678, 13.18447493468213, 13.61037165421069, 14.27730521318330, 14.89933853811091, 15.43871991917556, 15.97191325793299, 16.81232642868411
