L(s) = 1 | − 5.18·2-s + 18.9·4-s + 7·7-s − 56.5·8-s + 35.9·11-s + 45.2·13-s − 36.3·14-s + 142.·16-s − 113.·17-s + 61.5·19-s − 186.·22-s + 30.6·23-s − 234.·26-s + 132.·28-s − 214.·29-s + 164.·31-s − 284.·32-s + 586.·34-s − 410.·37-s − 319.·38-s + 309.·41-s − 29.9·43-s + 679.·44-s − 158.·46-s − 483.·47-s + 49·49-s + 855.·52-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 2.36·4-s + 0.377·7-s − 2.49·8-s + 0.985·11-s + 0.965·13-s − 0.693·14-s + 2.21·16-s − 1.61·17-s + 0.743·19-s − 1.80·22-s + 0.277·23-s − 1.77·26-s + 0.892·28-s − 1.37·29-s + 0.951·31-s − 1.57·32-s + 2.96·34-s − 1.82·37-s − 1.36·38-s + 1.17·41-s − 0.106·43-s + 2.32·44-s − 0.508·46-s − 1.50·47-s + 0.142·49-s + 2.28·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
good | 2 | \( 1 + 5.18T + 8T^{2} \) |
| 11 | \( 1 - 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 29.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 151.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 89.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 714.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 297.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 492.T + 9.12e5T^{2} \) |
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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
−8.921502478667356202632500361430, −8.101825320904199407378471550997, −7.19921680547288301051593757279, −6.62466394172784678154635173273, −5.78449714296418674283138759688, −4.35555791140189343006475750723, −3.16113950594293333732509025440, −1.90190659764656179828158986516, −1.22691046867859173709849201728, 0,
1.22691046867859173709849201728, 1.90190659764656179828158986516, 3.16113950594293333732509025440, 4.35555791140189343006475750723, 5.78449714296418674283138759688, 6.62466394172784678154635173273, 7.19921680547288301051593757279, 8.101825320904199407378471550997, 8.921502478667356202632500361430