| L(s) = 1 | + 5-s − 2·11-s − 13-s + 17-s − 8·19-s − 4·25-s + 5·29-s + 3·31-s − 7·41-s + 4·43-s − 9·47-s − 2·55-s + 3·59-s − 4·61-s − 65-s + 8·67-s + 6·73-s − 8·79-s + 9·83-s + 85-s − 2·89-s − 8·95-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.277·13-s + 0.242·17-s − 1.83·19-s − 4/5·25-s + 0.928·29-s + 0.538·31-s − 1.09·41-s + 0.609·43-s − 1.31·47-s − 0.269·55-s + 0.390·59-s − 0.512·61-s − 0.124·65-s + 0.977·67-s + 0.702·73-s − 0.900·79-s + 0.987·83-s + 0.108·85-s − 0.211·89-s − 0.820·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.69036240084726, −13.38378449462258, −12.77628874272344, −12.52969051915153, −11.85623639557020, −11.43604560240764, −10.78205379201427, −10.37837952063040, −9.954517676325810, −9.571261329703721, −8.763810355490606, −8.453568990491676, −7.954829571068404, −7.409728220738087, −6.705080461233493, −6.304741948088964, −5.885257780058624, −5.109797687687041, −4.762580328266321, −4.143333482882106, −3.475116626065172, −2.823730679840188, −2.160953188944057, −1.807871927313173, −0.7862263224898604, 0,
0.7862263224898604, 1.807871927313173, 2.160953188944057, 2.823730679840188, 3.475116626065172, 4.143333482882106, 4.762580328266321, 5.109797687687041, 5.885257780058624, 6.304741948088964, 6.705080461233493, 7.409728220738087, 7.954829571068404, 8.453568990491676, 8.763810355490606, 9.571261329703721, 9.954517676325810, 10.37837952063040, 10.78205379201427, 11.43604560240764, 11.85623639557020, 12.52969051915153, 12.77628874272344, 13.38378449462258, 13.69036240084726
