| L(s) = 1 | + 2·3-s − 4-s + 2·5-s − 2·7-s + 3·9-s + 6·11-s − 2·12-s − 2·13-s + 4·15-s − 3·16-s + 2·19-s − 2·20-s − 4·21-s + 3·25-s + 4·27-s + 2·28-s + 6·29-s + 4·31-s + 12·33-s − 4·35-s − 3·36-s − 2·37-s − 4·39-s + 6·41-s − 2·43-s − 6·44-s + 6·45-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.755·7-s + 9-s + 1.80·11-s − 0.577·12-s − 0.554·13-s + 1.03·15-s − 3/4·16-s + 0.458·19-s − 0.447·20-s − 0.872·21-s + 3/5·25-s + 0.769·27-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 2.08·33-s − 0.676·35-s − 1/2·36-s − 0.328·37-s − 0.640·39-s + 0.937·41-s − 0.304·43-s − 0.904·44-s + 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.322585834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322585834\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 256 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 168 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−12.15146469924703303495839368295, −11.66631361450594969104298348504, −11.16406106427020001019077922362, −10.40111776022432643504222121874, −9.812863144240357561180810209366, −9.726442139148524098460702342354, −9.102257630028187947415108993731, −9.064337119716004837733582986763, −8.357492743998930486544211870736, −7.87435648646176746695740198796, −7.03591498952022486703765867586, −6.67728330568751329913910617748, −6.33842567697968775115131274888, −5.58359033619848602859785877093, −4.58467533642185988339614222074, −4.46417869343420470917120240068, −3.49393018214745799274320542749, −3.01708959245831383050013519549, −2.19944547378500630649591783378, −1.27918120443323915706052052566,
1.27918120443323915706052052566, 2.19944547378500630649591783378, 3.01708959245831383050013519549, 3.49393018214745799274320542749, 4.46417869343420470917120240068, 4.58467533642185988339614222074, 5.58359033619848602859785877093, 6.33842567697968775115131274888, 6.67728330568751329913910617748, 7.03591498952022486703765867586, 7.87435648646176746695740198796, 8.357492743998930486544211870736, 9.064337119716004837733582986763, 9.102257630028187947415108993731, 9.726442139148524098460702342354, 9.812863144240357561180810209366, 10.40111776022432643504222121874, 11.16406106427020001019077922362, 11.66631361450594969104298348504, 12.15146469924703303495839368295
