| L(s) = 1 | + 2·3-s − 3·4-s + 2·5-s + 9-s − 6·12-s + 4·15-s + 5·16-s − 6·20-s + 4·23-s − 25-s − 4·27-s + 2·31-s − 3·36-s + 12·37-s + 2·45-s + 2·47-s + 10·48-s + 8·49-s + 6·53-s − 12·60-s − 3·64-s + 6·67-s + 8·69-s − 6·71-s − 2·75-s + 10·80-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s + 0.894·5-s + 1/3·9-s − 1.73·12-s + 1.03·15-s + 5/4·16-s − 1.34·20-s + 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.359·31-s − 1/2·36-s + 1.97·37-s + 0.298·45-s + 0.291·47-s + 1.44·48-s + 8/7·49-s + 0.824·53-s − 1.54·60-s − 3/8·64-s + 0.733·67-s + 0.963·69-s − 0.712·71-s − 0.230·75-s + 1.11·80-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.763312357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.763312357\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−7.61903597546549763610329784051, −7.31746964320638785746059580449, −6.69367836026985911925712888957, −6.08630065779101110146645115760, −5.87199295441569638505379699624, −5.37324179889398776145644655774, −4.89847213809135812830557412675, −4.51999187028564821606503198690, −3.98973302902399015605437597128, −3.67532222647555978345005795222, −3.07373473851789991663251978316, −2.50499289404960237862393568019, −2.19657804181307391051180413721, −1.30271825511364110114353963513, −0.63046087289744001639918732362,
0.63046087289744001639918732362, 1.30271825511364110114353963513, 2.19657804181307391051180413721, 2.50499289404960237862393568019, 3.07373473851789991663251978316, 3.67532222647555978345005795222, 3.98973302902399015605437597128, 4.51999187028564821606503198690, 4.89847213809135812830557412675, 5.37324179889398776145644655774, 5.87199295441569638505379699624, 6.08630065779101110146645115760, 6.69367836026985911925712888957, 7.31746964320638785746059580449, 7.61903597546549763610329784051
