| L(s) = 1 | − 2-s − 4-s − 2·5-s + 7-s + 3·8-s + 2·10-s − 11-s − 14-s − 16-s − 4·19-s + 2·20-s + 22-s + 4·23-s − 25-s − 28-s − 2·29-s + 10·31-s − 5·32-s − 2·35-s − 4·37-s + 4·38-s − 6·40-s − 10·41-s − 2·43-s + 44-s − 4·46-s + 10·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s + 1.06·8-s + 0.632·10-s − 0.301·11-s − 0.267·14-s − 1/4·16-s − 0.917·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s − 0.188·28-s − 0.371·29-s + 1.79·31-s − 0.883·32-s − 0.338·35-s − 0.657·37-s + 0.648·38-s − 0.948·40-s − 1.56·41-s − 0.304·43-s + 0.150·44-s − 0.589·46-s + 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.95111075272478, −13.62036457599590, −13.14861524082475, −12.58757649721966, −12.07099832634742, −11.60395406200929, −11.12433074808038, −10.49256219027735, −10.28607931561868, −9.705785729807791, −8.924964153070776, −8.716438150623396, −8.244094921218404, −7.727839725576307, −7.358075395825117, −6.785187770742331, −6.091206724454164, −5.437348257087129, −4.724370439781052, −4.477713626021205, −3.934512696793181, −3.187139311897781, −2.616591914845203, −1.607458395857290, −1.221777878285528, 0, 0,
1.221777878285528, 1.607458395857290, 2.616591914845203, 3.187139311897781, 3.934512696793181, 4.477713626021205, 4.724370439781052, 5.437348257087129, 6.091206724454164, 6.785187770742331, 7.358075395825117, 7.727839725576307, 8.244094921218404, 8.716438150623396, 8.924964153070776, 9.705785729807791, 10.28607931561868, 10.49256219027735, 11.12433074808038, 11.60395406200929, 12.07099832634742, 12.58757649721966, 13.14861524082475, 13.62036457599590, 13.95111075272478
