| L(s) = 1 | + 2-s − 4-s − 3·8-s − 3·9-s + 4·11-s − 16-s − 3·18-s + 4·22-s + 8·23-s − 5·25-s + 2·29-s + 5·32-s + 3·36-s − 6·37-s − 12·43-s − 4·44-s + 8·46-s − 5·50-s − 10·53-s + 2·58-s + 7·64-s + 4·67-s + 16·71-s + 9·72-s − 6·74-s + 8·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s + 1.20·11-s − 1/4·16-s − 0.707·18-s + 0.852·22-s + 1.66·23-s − 25-s + 0.371·29-s + 0.883·32-s + 1/2·36-s − 0.986·37-s − 1.82·43-s − 0.603·44-s + 1.17·46-s − 0.707·50-s − 1.37·53-s + 0.262·58-s + 7/8·64-s + 0.488·67-s + 1.89·71-s + 1.06·72-s − 0.697·74-s + 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9666558528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9666558528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.25649719028727745654998030256, −14.36789003254179224985767994212, −13.55829126515041538892762617601, −12.27943344760252552732176729088, −11.30850265031875128052013945557, −9.489525085039792097687572288594, −8.498120181782134288657898916471, −6.47803659589426412986704519993, −5.08673463814783736975452386140, −3.45773984941604093394269964405,
3.45773984941604093394269964405, 5.08673463814783736975452386140, 6.47803659589426412986704519993, 8.498120181782134288657898916471, 9.489525085039792097687572288594, 11.30850265031875128052013945557, 12.27943344760252552732176729088, 13.55829126515041538892762617601, 14.36789003254179224985767994212, 15.25649719028727745654998030256
