| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s − 2·9-s + 10-s + 12-s + 6·13-s + 15-s − 16-s + 2·18-s + 20-s − 3·24-s + 25-s − 6·26-s + 5·27-s − 30-s + 4·31-s − 5·32-s + 2·36-s + 12·37-s − 6·39-s − 3·40-s − 9·41-s + 2·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s + 0.258·15-s − 1/4·16-s + 0.471·18-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.962·27-s − 0.182·30-s + 0.718·31-s − 0.883·32-s + 1/3·36-s + 1.97·37-s − 0.960·39-s − 0.474·40-s − 1.40·41-s + 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.193237577816539325786479251914, −8.008122378873668824712351767130, −7.33495921061671416443383382408, −6.77697985512635763964628431112, −6.23410951044089958662364809747, −6.03396514791083212928763933125, −5.38339623206612517682324917908, −4.82194693874908794745894928249, −4.46437757877781096400659631045, −3.84483334155876524672758322690, −3.30951788395041666119605453457, −2.71695677809996229410312034877, −1.59536810300571759951317107559, −0.998102833276964457173100277187, 0,
0.998102833276964457173100277187, 1.59536810300571759951317107559, 2.71695677809996229410312034877, 3.30951788395041666119605453457, 3.84483334155876524672758322690, 4.46437757877781096400659631045, 4.82194693874908794745894928249, 5.38339623206612517682324917908, 6.03396514791083212928763933125, 6.23410951044089958662364809747, 6.77697985512635763964628431112, 7.33495921061671416443383382408, 8.008122378873668824712351767130, 8.193237577816539325786479251914
