| L(s) = 1 | + 2·2-s − 4-s − 8·8-s + 9-s − 8·11-s − 7·16-s + 2·18-s − 16·22-s − 4·29-s + 14·32-s − 36-s + 20·37-s − 8·43-s + 8·44-s − 7·49-s + 20·53-s − 8·58-s + 35·64-s − 24·67-s − 16·71-s − 8·72-s + 40·74-s + 81-s − 16·86-s + 64·88-s − 14·98-s − 8·99-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s + 1/3·9-s − 2.41·11-s − 7/4·16-s + 0.471·18-s − 3.41·22-s − 0.742·29-s + 2.47·32-s − 1/6·36-s + 3.28·37-s − 1.21·43-s + 1.20·44-s − 49-s + 2.74·53-s − 1.05·58-s + 35/8·64-s − 2.93·67-s − 1.89·71-s − 0.942·72-s + 4.64·74-s + 1/9·81-s − 1.72·86-s + 6.82·88-s − 1.41·98-s − 0.804·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.352025311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.352025311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.843956595114680827452383410006, −8.461212909606671780489016897710, −7.78056789512660315702386216522, −7.68003844385765985083943135134, −6.92482379445403167319259433221, −6.00448844798074713207333092038, −5.86497653842965752491328535183, −5.44640444640032659769086760535, −4.79896242020124990959414772446, −4.52160444884874669934496061646, −4.11644977935212568952932430273, −3.13361812930802426100944548657, −3.01549784140686353642765162463, −2.20286953987450620546650146772, −0.53965130087831198857502799684,
0.53965130087831198857502799684, 2.20286953987450620546650146772, 3.01549784140686353642765162463, 3.13361812930802426100944548657, 4.11644977935212568952932430273, 4.52160444884874669934496061646, 4.79896242020124990959414772446, 5.44640444640032659769086760535, 5.86497653842965752491328535183, 6.00448844798074713207333092038, 6.92482379445403167319259433221, 7.68003844385765985083943135134, 7.78056789512660315702386216522, 8.461212909606671780489016897710, 8.843956595114680827452383410006
