| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s + 4·13-s − 15-s + 2·17-s − 6·19-s + 4·23-s + 25-s + 27-s − 10·29-s + 2·31-s − 2·33-s − 2·37-s + 4·39-s + 10·41-s − 4·43-s − 45-s − 8·47-s + 2·51-s + 4·53-s + 2·55-s − 6·57-s + 4·59-s + 2·61-s − 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s + 0.485·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.359·31-s − 0.348·33-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 0.280·51-s + 0.549·53-s + 0.269·55-s − 0.794·57-s + 0.520·59-s + 0.256·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.57845883822778, −15.16915688750162, −14.69010231833712, −14.25454592023709, −13.29241442185343, −13.15759611564274, −12.71018646407432, −11.89008728451098, −11.29817684241992, −10.76142639291699, −10.39453634595089, −9.527163715328356, −9.046489579170611, −8.422492545761078, −8.035769088409884, −7.402762141108314, −6.800511053030912, −6.085425634890318, −5.471052899590203, −4.691940179745022, −3.977661618427589, −3.508485046729758, −2.758619827976275, −1.982855661932923, −1.135011283035165, 0,
1.135011283035165, 1.982855661932923, 2.758619827976275, 3.508485046729758, 3.977661618427589, 4.691940179745022, 5.471052899590203, 6.085425634890318, 6.800511053030912, 7.402762141108314, 8.035769088409884, 8.422492545761078, 9.046489579170611, 9.527163715328356, 10.39453634595089, 10.76142639291699, 11.29817684241992, 11.89008728451098, 12.71018646407432, 13.15759611564274, 13.29241442185343, 14.25454592023709, 14.69010231833712, 15.16915688750162, 15.57845883822778
