| L(s) = 1 | − 4-s + 6·9-s − 8·11-s + 16-s + 8·19-s + 12·29-s − 8·31-s − 6·36-s − 12·41-s + 8·44-s + 14·49-s − 8·59-s + 20·61-s − 64-s − 8·76-s + 8·79-s + 27·81-s − 4·89-s − 48·99-s + 12·101-s − 36·109-s − 12·116-s + 26·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·9-s − 2.41·11-s + 1/4·16-s + 1.83·19-s + 2.22·29-s − 1.43·31-s − 36-s − 1.87·41-s + 1.20·44-s + 2·49-s − 1.04·59-s + 2.56·61-s − 1/8·64-s − 0.917·76-s + 0.900·79-s + 3·81-s − 0.423·89-s − 4.82·99-s + 1.19·101-s − 3.44·109-s − 1.11·116-s + 2.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.055130252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055130252\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−9.468383275079430406733088322109, −9.145902878468978505721214128673, −8.583453911494714143430840010870, −8.217564159640307019973250620605, −7.77494176285693375285206165599, −7.53063721972286753901596431189, −7.21849460602420205191808894740, −6.74158224590255811125427061575, −6.40953919155191372822760475135, −5.46480065850068405645596287034, −5.38479070187470452740184433098, −4.98216256622746945432233063758, −4.72757264560054026652267358390, −3.87851995725608663304788329540, −3.82612923650194385924698859474, −2.95897255745322882044362635179, −2.66604041419969725124253779400, −1.93306319380909118376459116832, −1.25159062099686705601434889498, −0.58709966963459215480189321494,
0.58709966963459215480189321494, 1.25159062099686705601434889498, 1.93306319380909118376459116832, 2.66604041419969725124253779400, 2.95897255745322882044362635179, 3.82612923650194385924698859474, 3.87851995725608663304788329540, 4.72757264560054026652267358390, 4.98216256622746945432233063758, 5.38479070187470452740184433098, 5.46480065850068405645596287034, 6.40953919155191372822760475135, 6.74158224590255811125427061575, 7.21849460602420205191808894740, 7.53063721972286753901596431189, 7.77494176285693375285206165599, 8.217564159640307019973250620605, 8.583453911494714143430840010870, 9.145902878468978505721214128673, 9.468383275079430406733088322109
