| L(s) = 1 | − 2·5-s − 2·9-s − 2·13-s + 3·25-s + 4·45-s − 2·49-s + 12·53-s − 8·61-s + 4·65-s − 5·81-s − 12·89-s + 24·97-s + 12·101-s + 24·109-s − 12·113-s + 4·117-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2/3·9-s − 0.554·13-s + 3/5·25-s + 0.596·45-s − 2/7·49-s + 1.64·53-s − 1.02·61-s + 0.496·65-s − 5/9·81-s − 1.27·89-s + 2.43·97-s + 1.19·101-s + 2.29·109-s − 1.12·113-s + 0.369·117-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−7.80019523012963931903075954936, −7.31280066535498996112985725664, −7.26927111972768690986858989426, −6.55596474111111648673289672386, −6.05877445646593256691786933704, −5.70242119179958153730749881149, −5.05114780090473010136074821908, −4.71246525005282268147051099678, −4.19328938157817020898215093129, −3.60778916273215204704984402016, −3.17849108100061118271871533883, −2.58295993891423193456421003265, −1.97520006840294894021148545522, −0.942726694253668839126892786514, 0,
0.942726694253668839126892786514, 1.97520006840294894021148545522, 2.58295993891423193456421003265, 3.17849108100061118271871533883, 3.60778916273215204704984402016, 4.19328938157817020898215093129, 4.71246525005282268147051099678, 5.05114780090473010136074821908, 5.70242119179958153730749881149, 6.05877445646593256691786933704, 6.55596474111111648673289672386, 7.26927111972768690986858989426, 7.31280066535498996112985725664, 7.80019523012963931903075954936
