| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s + 4·13-s + 16-s − 18-s − 2·19-s + 2·24-s − 5·25-s − 4·26-s + 4·27-s + 6·29-s − 4·31-s − 32-s + 36-s − 2·37-s + 2·38-s − 8·39-s + 6·41-s + 8·43-s + 12·47-s − 2·48-s + 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.408·24-s − 25-s − 0.784·26-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1/6·36-s − 0.328·37-s + 0.324·38-s − 1.28·39-s + 0.937·41-s + 1.21·43-s + 1.75·47-s − 0.288·48-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9720606617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9720606617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.52438442525064, −14.73656500758754, −14.19557060041075, −13.54484083780491, −12.99404196784882, −12.16852999619247, −12.06933226531708, −11.24977310470779, −10.93024025151654, −10.48927574534232, −9.944712601885258, −9.139429875069651, −8.727997990211820, −8.125592459516536, −7.451920801474087, −6.804542028509099, −6.279489933602623, −5.700185020196356, −5.390015840914639, −4.295396462186042, −3.886972406084437, −2.849335745149190, −2.114873981625971, −1.125790132487681, −0.5395601724337955,
0.5395601724337955, 1.125790132487681, 2.114873981625971, 2.849335745149190, 3.886972406084437, 4.295396462186042, 5.390015840914639, 5.700185020196356, 6.279489933602623, 6.804542028509099, 7.451920801474087, 8.125592459516536, 8.727997990211820, 9.139429875069651, 9.944712601885258, 10.48927574534232, 10.93024025151654, 11.24977310470779, 12.06933226531708, 12.16852999619247, 12.99404196784882, 13.54484083780491, 14.19557060041075, 14.73656500758754, 15.52438442525064
