| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 2·11-s + 5·16-s + 4·22-s − 16·23-s − 10·25-s − 4·29-s + 6·32-s + 4·37-s − 8·43-s + 6·44-s − 32·46-s − 20·50-s − 28·53-s − 8·58-s + 7·64-s + 8·67-s + 8·74-s − 32·79-s − 16·86-s + 8·88-s − 48·92-s − 30·100-s − 56·106-s − 24·107-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 0.603·11-s + 5/4·16-s + 0.852·22-s − 3.33·23-s − 2·25-s − 0.742·29-s + 1.06·32-s + 0.657·37-s − 1.21·43-s + 0.904·44-s − 4.71·46-s − 2.82·50-s − 3.84·53-s − 1.05·58-s + 7/8·64-s + 0.977·67-s + 0.929·74-s − 3.60·79-s − 1.72·86-s + 0.852·88-s − 5.00·92-s − 3·100-s − 5.43·106-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−7.35254590003840099969194456766, −7.26291126010817866608730322856, −6.45379480453288463630673707805, −6.45135567153230697369483542793, −6.04357401203699063151521472821, −5.91945577831179393514408992648, −5.44412349960301715849966381951, −5.18207569668082943879702524376, −4.57617347953927903099198520157, −4.31742103928807866958260303629, −3.94894170634214286509505704383, −3.88749025465319802027830176707, −3.23199485372711231632169723570, −3.09622179569473426352489015259, −2.28308644108255541101560857236, −2.12973997053598942616145080949, −1.53611668743012339964030891379, −1.45576040914551718071665578373, 0, 0,
1.45576040914551718071665578373, 1.53611668743012339964030891379, 2.12973997053598942616145080949, 2.28308644108255541101560857236, 3.09622179569473426352489015259, 3.23199485372711231632169723570, 3.88749025465319802027830176707, 3.94894170634214286509505704383, 4.31742103928807866958260303629, 4.57617347953927903099198520157, 5.18207569668082943879702524376, 5.44412349960301715849966381951, 5.91945577831179393514408992648, 6.04357401203699063151521472821, 6.45135567153230697369483542793, 6.45379480453288463630673707805, 7.26291126010817866608730322856, 7.35254590003840099969194456766
