| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 13-s + 15-s + 6·17-s − 8·19-s − 2·21-s − 8·23-s + 25-s − 27-s − 4·29-s − 2·31-s − 2·35-s − 8·37-s − 39-s + 6·41-s − 12·43-s − 45-s + 8·47-s − 3·49-s − 6·51-s + 12·53-s + 8·57-s + 8·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 1.45·17-s − 1.83·19-s − 0.436·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.359·31-s − 0.338·35-s − 1.31·37-s − 0.160·39-s + 0.937·41-s − 1.82·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.840·51-s + 1.64·53-s + 1.05·57-s + 1.04·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.36580307492332, −12.70568380577032, −12.32060302314036, −11.91362386776337, −11.61107794584379, −10.90809674197321, −10.63742503222095, −10.17884050016517, −9.722102886417170, −8.990579481828520, −8.410908176591098, −8.187675768456828, −7.616609070024523, −7.119092048067073, −6.597323385030936, −5.915135209021944, −5.622446550623924, −5.089238970745047, −4.395089833167840, −3.975923248871871, −3.597689456672884, −2.746864564328659, −1.859926709021654, −1.681378255108991, −0.6924589209824506, 0,
0.6924589209824506, 1.681378255108991, 1.859926709021654, 2.746864564328659, 3.597689456672884, 3.975923248871871, 4.395089833167840, 5.089238970745047, 5.622446550623924, 5.915135209021944, 6.597323385030936, 7.119092048067073, 7.616609070024523, 8.187675768456828, 8.410908176591098, 8.990579481828520, 9.722102886417170, 10.17884050016517, 10.63742503222095, 10.90809674197321, 11.61107794584379, 11.91362386776337, 12.32060302314036, 12.70568380577032, 13.36580307492332
