| L(s) = 1 | + 5-s + 7-s − 5·11-s + 2·17-s + 19-s + 23-s − 4·25-s + 4·29-s + 9·31-s + 35-s + 5·37-s − 9·41-s + 10·43-s − 6·47-s + 49-s + 12·53-s − 5·55-s + 14·59-s + 8·67-s + 13·71-s − 2·73-s − 5·77-s − 6·79-s + 4·83-s + 2·85-s − 9·89-s + 95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.50·11-s + 0.485·17-s + 0.229·19-s + 0.208·23-s − 4/5·25-s + 0.742·29-s + 1.61·31-s + 0.169·35-s + 0.821·37-s − 1.40·41-s + 1.52·43-s − 0.875·47-s + 1/7·49-s + 1.64·53-s − 0.674·55-s + 1.82·59-s + 0.977·67-s + 1.54·71-s − 0.234·73-s − 0.569·77-s − 0.675·79-s + 0.439·83-s + 0.216·85-s − 0.953·89-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944950498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944950498\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.478426573102990555594665044264, −8.098584445070679785979739673973, −7.30779100190829260650015981954, −6.41579425095070423565731763913, −5.51926476472048359134068258767, −5.05061740660147245068087666602, −4.05627857283460152298243124357, −2.88260927317063896946056802272, −2.20921145148586991820085196748, −0.853219031875925699013328984683,
0.853219031875925699013328984683, 2.20921145148586991820085196748, 2.88260927317063896946056802272, 4.05627857283460152298243124357, 5.05061740660147245068087666602, 5.51926476472048359134068258767, 6.41579425095070423565731763913, 7.30779100190829260650015981954, 8.098584445070679785979739673973, 8.478426573102990555594665044264
