L(s) = 1 | − 4.97·2-s − 3·3-s + 16.7·4-s + 14.9·6-s − 5.48·7-s − 43.3·8-s + 9·9-s + 11·11-s − 50.1·12-s − 24.5·13-s + 27.2·14-s + 81.6·16-s + 59.3·17-s − 44.7·18-s + 5.89·19-s + 16.4·21-s − 54.6·22-s + 68.4·23-s + 129.·24-s + 122.·26-s − 27·27-s − 91.6·28-s − 265.·29-s − 196.·31-s − 59.2·32-s − 33·33-s − 295.·34-s + ⋯ |
L(s) = 1 | − 1.75·2-s − 0.577·3-s + 2.08·4-s + 1.01·6-s − 0.296·7-s − 1.91·8-s + 0.333·9-s + 0.301·11-s − 1.20·12-s − 0.524·13-s + 0.520·14-s + 1.27·16-s + 0.847·17-s − 0.585·18-s + 0.0711·19-s + 0.170·21-s − 0.529·22-s + 0.620·23-s + 1.10·24-s + 0.921·26-s − 0.192·27-s − 0.618·28-s − 1.69·29-s − 1.13·31-s − 0.327·32-s − 0.174·33-s − 1.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 4.97T + 8T^{2} \) |
| 7 | \( 1 + 5.48T + 343T^{2} \) |
| 13 | \( 1 + 24.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.89T + 6.85e3T^{2} \) |
| 23 | \( 1 - 68.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 424.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 339.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 662.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 313.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 153.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 153.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 403.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 959.T + 9.12e5T^{2} \) |
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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
−9.455059253445627846431972477339, −8.789070329456716328700463922895, −7.59864198458999920060838161953, −7.22476365326484019209834724176, −6.19632369292971837073748416110, −5.25841283615790482793916750187, −3.65529333290895452112174093378, −2.24488153647382894389430809733, −1.09068984393183498145687599081, 0,
1.09068984393183498145687599081, 2.24488153647382894389430809733, 3.65529333290895452112174093378, 5.25841283615790482793916750187, 6.19632369292971837073748416110, 7.22476365326484019209834724176, 7.59864198458999920060838161953, 8.789070329456716328700463922895, 9.455059253445627846431972477339