| L(s) = 1 | + 2·5-s − 5·9-s − 2·13-s − 6·17-s + 3·25-s − 18·29-s − 20·37-s − 10·45-s + 49-s + 16·61-s − 4·65-s + 28·73-s + 16·81-s − 12·85-s + 24·89-s + 34·97-s − 12·101-s − 38·109-s − 12·113-s + 10·117-s − 13·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 36·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 5/3·9-s − 0.554·13-s − 1.45·17-s + 3/5·25-s − 3.34·29-s − 3.28·37-s − 1.49·45-s + 1/7·49-s + 2.04·61-s − 0.496·65-s + 3.27·73-s + 16/9·81-s − 1.30·85-s + 2.54·89-s + 3.45·97-s − 1.19·101-s − 3.63·109-s − 1.12·113-s + 0.924·117-s − 1.18·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.98·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−9.375175591104324316941268909160, −8.980861688416760643046869733716, −8.753603075201557578492983990278, −8.046356379591346049691472796650, −7.50080687239149741983670353733, −6.67050874393051432711601966688, −6.60075882433919903374418695456, −5.59805653954512089440221213789, −5.40018100386713622724414679311, −4.97734488095803542825402310226, −3.80030272906745320072379387841, −3.40261805032806216019286027323, −2.16340813030352361640823678027, −2.14587146010097196258543648767, 0,
2.14587146010097196258543648767, 2.16340813030352361640823678027, 3.40261805032806216019286027323, 3.80030272906745320072379387841, 4.97734488095803542825402310226, 5.40018100386713622724414679311, 5.59805653954512089440221213789, 6.60075882433919903374418695456, 6.67050874393051432711601966688, 7.50080687239149741983670353733, 8.046356379591346049691472796650, 8.753603075201557578492983990278, 8.980861688416760643046869733716, 9.375175591104324316941268909160
