| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 2·11-s + 5·16-s + 4·22-s + 8·23-s − 8·25-s + 12·29-s − 6·32-s − 16·37-s − 16·43-s − 6·44-s − 16·46-s + 16·50-s − 24·58-s + 7·64-s − 16·67-s + 16·71-s + 32·74-s + 8·79-s + 32·86-s + 8·88-s + 24·92-s − 24·100-s − 16·107-s + 8·109-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 + 1.66·23-s − 8/5·25-s + 2.22·29-s − 1.06·32-s − 2.63·37-s − 2.43·43-s − 0.904·44-s − 2.35·46-s + 2.26·50-s − 3.15·58-s + 7/8·64-s − 1.95·67-s + 1.89·71-s + 3.71·74-s + 0.900·79-s + 3.45·86-s + 0.852·88-s + 2.50·92-s − 2.39·100-s − 1.54·107-s + 0.766·109-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 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
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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
−7.44101026525792074041023290162, −7.28221579696518488307160152922, −6.83241933588875437708540172818, −6.76647536949665559531313365175, −6.15855033748627495505040230747, −6.11686917129522057351557448824, −5.35074155741522510869068543679, −5.21118544326895154803339956925, −4.85308670455194117520888050716, −4.52228542407737730630259468249, −3.69952845422182766797975115257, −3.58566315054123885461219146000, −2.99428346365492788390692356300, −2.86985277158359714171875405724, −2.06572600660909889910839720824, −2.04319898960403537319275019497, −1.23854838287504997368745280975, −1.06399021800454625787883543368, 0, 0,
1.06399021800454625787883543368, 1.23854838287504997368745280975, 2.04319898960403537319275019497, 2.06572600660909889910839720824, 2.86985277158359714171875405724, 2.99428346365492788390692356300, 3.58566315054123885461219146000, 3.69952845422182766797975115257, 4.52228542407737730630259468249, 4.85308670455194117520888050716, 5.21118544326895154803339956925, 5.35074155741522510869068543679, 6.11686917129522057351557448824, 6.15855033748627495505040230747, 6.76647536949665559531313365175, 6.83241933588875437708540172818, 7.28221579696518488307160152922, 7.44101026525792074041023290162
