| L(s) = 1 | − 2·2-s + 2·4-s − 3·5-s + 6·10-s − 2·11-s + 6·13-s − 4·16-s − 3·17-s − 6·19-s − 6·20-s + 4·22-s + 8·23-s + 4·25-s − 12·26-s + 2·29-s + 6·31-s + 8·32-s + 6·34-s + 9·37-s + 12·38-s − 9·41-s − 9·43-s − 4·44-s − 16·46-s + 3·47-s − 8·50-s + 12·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.34·5-s + 1.89·10-s − 0.603·11-s + 1.66·13-s − 16-s − 0.727·17-s − 1.37·19-s − 1.34·20-s + 0.852·22-s + 1.66·23-s + 4/5·25-s − 2.35·26-s + 0.371·29-s + 1.07·31-s + 1.41·32-s + 1.02·34-s + 1.47·37-s + 1.94·38-s − 1.40·41-s − 1.37·43-s − 0.603·44-s − 2.35·46-s + 0.437·47-s − 1.13·50-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.890558367731343657232563624896, −8.434922105629152341884909496578, −8.006563649132861729931234193831, −6.98608566084697406804010635220, −6.36927615113021464171552256139, −4.79566875778408407423427665927, −4.00349709866533381716669420101, −2.79405220132507070327051254291, −1.26210148843102243348588568616, 0,
1.26210148843102243348588568616, 2.79405220132507070327051254291, 4.00349709866533381716669420101, 4.79566875778408407423427665927, 6.36927615113021464171552256139, 6.98608566084697406804010635220, 8.006563649132861729931234193831, 8.434922105629152341884909496578, 8.890558367731343657232563624896
