| L(s) = 1 | + 2·5-s − 2·9-s − 2·13-s − 12·17-s + 3·25-s + 12·29-s + 12·37-s − 4·45-s − 14·49-s + 4·61-s − 4·65-s + 12·73-s − 5·81-s − 24·85-s + 12·97-s − 12·101-s − 12·109-s + 12·113-s + 4·117-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2/3·9-s − 0.554·13-s − 2.91·17-s + 3/5·25-s + 2.22·29-s + 1.97·37-s − 0.596·45-s − 2·49-s + 0.512·61-s − 0.496·65-s + 1.40·73-s − 5/9·81-s − 2.60·85-s + 1.21·97-s − 1.19·101-s − 1.14·109-s + 1.12·113-s + 0.369·117-s − 0.181·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.966862383928881440655003735798, −7.39561199172765762009000013654, −6.73995582562381215551096178888, −6.46742372621991259230744798085, −6.32109088978854006350933074639, −5.71507515611670657325866245969, −5.03296562316088723460478207873, −4.68458332219322885087778524106, −4.43452672417510618546752868321, −3.66150609202387737964609502783, −2.80671211665565191776068265997, −2.51085196410167359218461980453, −2.11080508643635329880349152676, −1.13162970259400932412634941868, 0,
1.13162970259400932412634941868, 2.11080508643635329880349152676, 2.51085196410167359218461980453, 2.80671211665565191776068265997, 3.66150609202387737964609502783, 4.43452672417510618546752868321, 4.68458332219322885087778524106, 5.03296562316088723460478207873, 5.71507515611670657325866245969, 6.32109088978854006350933074639, 6.46742372621991259230744798085, 6.73995582562381215551096178888, 7.39561199172765762009000013654, 7.966862383928881440655003735798
