| L(s) = 1 | − 4·3-s + 3·4-s + 6·9-s − 12·12-s + 5·16-s − 4·17-s + 12·23-s − 25-s + 4·27-s + 4·29-s + 18·36-s − 20·43-s − 20·48-s − 2·49-s + 16·51-s + 4·53-s + 4·61-s + 3·64-s − 12·68-s − 48·69-s + 4·75-s − 24·79-s − 37·81-s − 16·87-s + 36·92-s − 3·100-s + 36·101-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3/2·4-s + 2·9-s − 3.46·12-s + 5/4·16-s − 0.970·17-s + 2.50·23-s − 1/5·25-s + 0.769·27-s + 0.742·29-s + 3·36-s − 3.04·43-s − 2.88·48-s − 2/7·49-s + 2.24·51-s + 0.549·53-s + 0.512·61-s + 3/8·64-s − 1.45·68-s − 5.77·69-s + 0.461·75-s − 2.70·79-s − 4.11·81-s − 1.71·87-s + 3.75·92-s − 0.299·100-s + 3.58·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9945309067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9945309067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.58405919083671902495924270829, −10.11895563034197909407823571774, −10.05238995310067270280235099912, −9.122425594493307535497078315070, −8.523513611570280965040314612522, −8.513704216344012009473508944188, −7.52598763457295311650776886095, −7.04618097845057008170066218918, −6.86819471427541184099415582226, −6.42307203504027513831984521348, −6.24420123868535353947613516284, −5.52795626289826992815154763643, −5.35083156401784355537659123724, −4.69448646887214829209010013656, −4.51406516314444314608791059779, −3.15812983439305247103446331676, −3.11385963893646510621928348170, −2.17283434610101842597137568418, −1.40403385772140644156874796999, −0.58015957265179580788170889107,
0.58015957265179580788170889107, 1.40403385772140644156874796999, 2.17283434610101842597137568418, 3.11385963893646510621928348170, 3.15812983439305247103446331676, 4.51406516314444314608791059779, 4.69448646887214829209010013656, 5.35083156401784355537659123724, 5.52795626289826992815154763643, 6.24420123868535353947613516284, 6.42307203504027513831984521348, 6.86819471427541184099415582226, 7.04618097845057008170066218918, 7.52598763457295311650776886095, 8.513704216344012009473508944188, 8.523513611570280965040314612522, 9.122425594493307535497078315070, 10.05238995310067270280235099912, 10.11895563034197909407823571774, 10.58405919083671902495924270829
