| L(s) = 1 | + 4·3-s − 2·5-s + 6·9-s − 8·15-s − 12·17-s + 4·19-s + 3·25-s − 4·27-s + 8·31-s − 12·45-s − 10·49-s − 48·51-s + 16·57-s − 24·59-s + 4·61-s − 4·67-s + 24·71-s + 4·73-s + 12·75-s − 16·79-s − 37·81-s + 24·85-s + 32·93-s − 8·95-s + 12·101-s − 28·103-s + 12·107-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 2·9-s − 2.06·15-s − 2.91·17-s + 0.917·19-s + 3/5·25-s − 0.769·27-s + 1.43·31-s − 1.78·45-s − 1.42·49-s − 6.72·51-s + 2.11·57-s − 3.12·59-s + 0.512·61-s − 0.488·67-s + 2.84·71-s + 0.468·73-s + 1.38·75-s − 1.80·79-s − 4.11·81-s + 2.60·85-s + 3.31·93-s − 0.820·95-s + 1.19·101-s − 2.75·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.347343430009299439276351390943, −7.71273110823499542846308181544, −7.70342498742006188501994585256, −7.06215723111114144957068885167, −6.39347529836665585887623697697, −6.27087624192875571051851265258, −5.14081916011398824138614106106, −4.74913960233451073492131590460, −4.12250433368686324236236368171, −3.82183399469630822167941819328, −3.12234796120650330097153328391, −2.76929890617261215013507568311, −2.31822883421800731675688556606, −1.57301709011122909909496546092, 0,
1.57301709011122909909496546092, 2.31822883421800731675688556606, 2.76929890617261215013507568311, 3.12234796120650330097153328391, 3.82183399469630822167941819328, 4.12250433368686324236236368171, 4.74913960233451073492131590460, 5.14081916011398824138614106106, 6.27087624192875571051851265258, 6.39347529836665585887623697697, 7.06215723111114144957068885167, 7.70342498742006188501994585256, 7.71273110823499542846308181544, 8.347343430009299439276351390943
