L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 15.8·5-s − 6·6-s − 8·8-s + 9·9-s + 31.7·10-s + 57.3·11-s + 12·12-s + 5.69·13-s − 47.6·15-s + 16·16-s − 51.8·17-s − 18·18-s + 16.2·19-s − 63.5·20-s − 114.·22-s − 213.·23-s − 24·24-s + 127.·25-s − 11.3·26-s + 27·27-s − 218.·29-s + 95.3·30-s − 251.·31-s − 32·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.42·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.00·10-s + 1.57·11-s + 0.288·12-s + 0.121·13-s − 0.821·15-s + 0.250·16-s − 0.740·17-s − 0.235·18-s + 0.195·19-s − 0.711·20-s − 1.11·22-s − 1.93·23-s − 0.204·24-s + 1.02·25-s − 0.0859·26-s + 0.192·27-s − 1.39·29-s + 0.580·30-s − 1.45·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 15.8T + 125T^{2} \) |
| 11 | \( 1 - 57.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.69T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 213.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 386.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 37.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 836.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 165.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 465.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 343.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.50e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 341.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 865.T + 9.12e5T^{2} \) |
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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
−11.01078268999428782279105875749, −9.651312137416242421496265502210, −8.926503701804876290893032428766, −8.010110502586491936596428866848, −7.30690027901813792296692594568, −6.21986903775267006160769438229, −4.23042052409539446516930576079, −3.52960274074151958935121362167, −1.74112416410043549236673096231, 0,
1.74112416410043549236673096231, 3.52960274074151958935121362167, 4.23042052409539446516930576079, 6.21986903775267006160769438229, 7.30690027901813792296692594568, 8.010110502586491936596428866848, 8.926503701804876290893032428766, 9.651312137416242421496265502210, 11.01078268999428782279105875749