| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s − 16·17-s − 12·23-s − 25-s − 4·27-s − 8·29-s − 3·36-s + 8·43-s − 2·48-s + 10·49-s + 32·51-s − 20·53-s − 20·61-s − 64-s + 16·68-s + 24·69-s + 2·75-s + 16·79-s + 5·81-s + 16·87-s + 12·92-s + 100-s + 32·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s − 3.88·17-s − 2.50·23-s − 1/5·25-s − 0.769·27-s − 1.48·29-s − 1/2·36-s + 1.21·43-s − 0.288·48-s + 10/7·49-s + 4.48·51-s − 2.74·53-s − 2.56·61-s − 1/8·64-s + 1.94·68-s + 2.88·69-s + 0.230·75-s + 1.80·79-s + 5/9·81-s + 1.71·87-s + 1.25·92-s + 1/10·100-s + 3.18·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−7.890723838570164006515342968873, −7.69524363933831705678297303833, −7.28114190331287182662159880881, −7.00385176977187641524415900809, −6.33117236701521112396022104275, −6.23406995952728775956924325643, −5.91826796095453046104039957369, −5.79640421327515474280345611260, −4.82102169894939032429475651892, −4.77144264386310586897385757542, −4.35644715598951209367070405300, −4.30093706890926341643824962610, −3.51802980583544664536812234277, −3.36633959734645076296051979797, −2.19284819031003011480493001001, −2.08811721803671611353124900959, −1.90945856466375441765065330650, −0.803750271251906704292505362934, 0, 0,
0.803750271251906704292505362934, 1.90945856466375441765065330650, 2.08811721803671611353124900959, 2.19284819031003011480493001001, 3.36633959734645076296051979797, 3.51802980583544664536812234277, 4.30093706890926341643824962610, 4.35644715598951209367070405300, 4.77144264386310586897385757542, 4.82102169894939032429475651892, 5.79640421327515474280345611260, 5.91826796095453046104039957369, 6.23406995952728775956924325643, 6.33117236701521112396022104275, 7.00385176977187641524415900809, 7.28114190331287182662159880881, 7.69524363933831705678297303833, 7.890723838570164006515342968873
