| L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s − 11-s + 6·13-s − 2·15-s − 6·17-s − 8·19-s − 4·21-s − 25-s − 27-s − 6·29-s + 33-s + 8·35-s − 6·37-s − 6·39-s − 10·41-s − 8·43-s + 2·45-s + 9·49-s + 6·51-s − 6·53-s − 2·55-s + 8·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.516·15-s − 1.45·17-s − 1.83·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s + 1.35·35-s − 0.986·37-s − 0.960·39-s − 1.56·41-s − 1.21·43-s + 0.298·45-s + 9/7·49-s + 0.840·51-s − 0.824·53-s − 0.269·55-s + 1.05·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.282924037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282924037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.98898450067062, −12.26870504147275, −11.53037805820554, −11.42775497671816, −10.88728720208484, −10.55057041262836, −10.27930085866568, −9.489978354551701, −8.901082473527067, −8.562285910456802, −8.287213575986531, −7.665902810384002, −6.973259990456580, −6.426538996011350, −6.238631809589230, −5.615202441342085, −5.089772521529660, −4.761246485665673, −4.068921527131964, −3.782103258234728, −2.841504614267137, −2.001003348823964, −1.725299019594003, −1.476007879869871, −0.2895983160599240,
0.2895983160599240, 1.476007879869871, 1.725299019594003, 2.001003348823964, 2.841504614267137, 3.782103258234728, 4.068921527131964, 4.761246485665673, 5.089772521529660, 5.615202441342085, 6.238631809589230, 6.426538996011350, 6.973259990456580, 7.665902810384002, 8.287213575986531, 8.562285910456802, 8.901082473527067, 9.489978354551701, 10.27930085866568, 10.55057041262836, 10.88728720208484, 11.42775497671816, 11.53037805820554, 12.26870504147275, 12.98898450067062
