| L(s) = 1 | + 2-s + 3-s + 4-s − 0.753·5-s + 6-s − 4.69·7-s + 8-s + 9-s − 0.753·10-s − 5.51·11-s + 12-s − 4.13·13-s − 4.69·14-s − 0.753·15-s + 16-s + 4.34·17-s + 18-s + 5.65·19-s − 0.753·20-s − 4.69·21-s − 5.51·22-s − 5.69·23-s + 24-s − 4.43·25-s − 4.13·26-s + 27-s − 4.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.336·5-s + 0.408·6-s − 1.77·7-s + 0.353·8-s + 0.333·9-s − 0.238·10-s − 1.66·11-s + 0.288·12-s − 1.14·13-s − 1.25·14-s − 0.194·15-s + 0.250·16-s + 1.05·17-s + 0.235·18-s + 1.29·19-s − 0.168·20-s − 1.02·21-s − 1.17·22-s − 1.18·23-s + 0.204·24-s − 0.886·25-s − 0.811·26-s + 0.192·27-s − 0.886·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 0.753T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 + 0.417T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 41 | \( 1 + 9.36T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 - 6.46T + 79T^{2} \) |
| 83 | \( 1 - 2.93T + 83T^{2} \) |
| 89 | \( 1 + 4.59T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−9.567304693105465161281847061193, −8.114976407795153467770668087375, −7.54750257708547906537849086359, −6.83632483959898722439332566181, −5.69450475349190432866064007291, −5.08256965940310810458814162038, −3.66274583474964439446445294686, −3.18437186873917280734313608602, −2.27842930446747794927817682402, 0,
2.27842930446747794927817682402, 3.18437186873917280734313608602, 3.66274583474964439446445294686, 5.08256965940310810458814162038, 5.69450475349190432866064007291, 6.83632483959898722439332566181, 7.54750257708547906537849086359, 8.114976407795153467770668087375, 9.567304693105465161281847061193
