| L(s) = 1 | − 2.64·2-s − 6.66·3-s − 25.0·4-s + 17.6·6-s + 12.8·7-s + 150.·8-s − 198.·9-s − 121·11-s + 166.·12-s + 485.·13-s − 34.0·14-s + 402.·16-s − 266.·17-s + 524.·18-s − 149.·19-s − 85.8·21-s + 319.·22-s + 3.21e3·23-s − 1.00e3·24-s − 1.28e3·26-s + 2.94e3·27-s − 322.·28-s + 2.94e3·29-s + 2.14e3·31-s − 5.88e3·32-s + 806.·33-s + 704.·34-s + ⋯ |
| L(s) = 1 | − 0.467·2-s − 0.427·3-s − 0.781·4-s + 0.199·6-s + 0.0993·7-s + 0.832·8-s − 0.817·9-s − 0.301·11-s + 0.334·12-s + 0.796·13-s − 0.0464·14-s + 0.393·16-s − 0.223·17-s + 0.381·18-s − 0.0951·19-s − 0.0424·21-s + 0.140·22-s + 1.26·23-s − 0.355·24-s − 0.371·26-s + 0.776·27-s − 0.0776·28-s + 0.651·29-s + 0.401·31-s − 1.01·32-s + 0.128·33-s + 0.104·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 2.64T + 32T^{2} \) |
| 3 | \( 1 + 6.66T + 243T^{2} \) |
| 7 | \( 1 - 12.8T + 1.68e4T^{2} \) |
| 13 | \( 1 - 485.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 266.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 149.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.21e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 808.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.50e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.59e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.27e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.32e5T + 8.58e9T^{2} \) |
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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
−10.66576474446181905040823150338, −9.498200386965088124288279522326, −8.661158953277657204260983590799, −7.928740972649067378390008679363, −6.53911096582029257658534386788, −5.42296623353871580112326065211, −4.48410805803587620188823896205, −3.03975266628050396419813732808, −1.18734961991201355729757522437, 0,
1.18734961991201355729757522437, 3.03975266628050396419813732808, 4.48410805803587620188823896205, 5.42296623353871580112326065211, 6.53911096582029257658534386788, 7.928740972649067378390008679363, 8.661158953277657204260983590799, 9.498200386965088124288279522326, 10.66576474446181905040823150338
