| L(s) = 1 | − 2·2-s + 2·4-s − 3·5-s − 2·7-s + 6·10-s − 5·11-s − 13-s + 4·14-s − 4·16-s − 5·19-s − 6·20-s + 10·22-s − 23-s + 4·25-s + 2·26-s − 4·28-s − 6·29-s − 10·31-s + 8·32-s + 6·35-s + 2·37-s + 10·38-s − 5·41-s + 43-s − 10·44-s + 2·46-s + 2·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.34·5-s − 0.755·7-s + 1.89·10-s − 1.50·11-s − 0.277·13-s + 1.06·14-s − 16-s − 1.14·19-s − 1.34·20-s + 2.13·22-s − 0.208·23-s + 4/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s − 1.79·31-s + 1.41·32-s + 1.01·35-s + 0.328·37-s + 1.62·38-s − 0.780·41-s + 0.152·43-s − 1.50·44-s + 0.294·46-s + 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.081022723276236914276753957321, −7.58408448077032956961751275797, −7.08910817117822589592710201732, −6.01244352681914030219761708009, −4.88598597278142628599449895472, −3.97459470051021210954332363879, −2.99831965891202149440095898837, −1.89532895980032283732016735838, 0, 0,
1.89532895980032283732016735838, 2.99831965891202149440095898837, 3.97459470051021210954332363879, 4.88598597278142628599449895472, 6.01244352681914030219761708009, 7.08910817117822589592710201732, 7.58408448077032956961751275797, 8.081022723276236914276753957321
