| L(s) = 1 | + 1.19·2-s − 6.56·4-s + 7·7-s − 17.4·8-s + 4.52·11-s − 14.5·13-s + 8.37·14-s + 31.7·16-s − 15.5·17-s − 14.8·19-s + 5.41·22-s + 166.·23-s − 17.4·26-s − 45.9·28-s − 194.·29-s + 104.·31-s + 177.·32-s − 18.6·34-s + 47.5·37-s − 17.7·38-s + 378.·41-s − 194.·43-s − 29.7·44-s + 199.·46-s − 488.·47-s + 49·49-s + 95.7·52-s + ⋯ |
| L(s) = 1 | + 0.422·2-s − 0.821·4-s + 0.377·7-s − 0.770·8-s + 0.124·11-s − 0.310·13-s + 0.159·14-s + 0.495·16-s − 0.221·17-s − 0.179·19-s + 0.0524·22-s + 1.51·23-s − 0.131·26-s − 0.310·28-s − 1.24·29-s + 0.605·31-s + 0.979·32-s − 0.0938·34-s + 0.211·37-s − 0.0758·38-s + 1.44·41-s − 0.691·43-s − 0.101·44-s + 0.639·46-s − 1.51·47-s + 0.142·49-s + 0.255·52-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 | \( 1 - 1.19T + 8T^{2} \) |
| 11 | \( 1 - 4.52T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 47.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 378.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 115.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 434.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 464.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 290.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 578.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 937.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 839.T + 9.12e5T^{2} \) |
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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
−8.734316066345159472129925836570, −7.947889862792966487865974296733, −7.03445063287397296067578856948, −6.05631413059383073384896999307, −5.17004850143888157817590030889, −4.56563497805090906697797444809, −3.64966226670344065188603638656, −2.65622382088595926026509903834, −1.25115933733093710893864531613, 0,
1.25115933733093710893864531613, 2.65622382088595926026509903834, 3.64966226670344065188603638656, 4.56563497805090906697797444809, 5.17004850143888157817590030889, 6.05631413059383073384896999307, 7.03445063287397296067578856948, 7.947889862792966487865974296733, 8.734316066345159472129925836570
