| L(s) = 1 | + 4-s + 2·5-s + 9-s + 4·11-s − 3·16-s + 2·20-s − 8·23-s + 3·25-s + 36-s − 4·37-s + 4·44-s + 2·45-s − 8·47-s + 2·49-s + 12·53-s + 8·55-s − 8·59-s − 7·64-s − 16·67-s − 6·80-s + 81-s + 20·89-s − 8·92-s + 4·97-s + 4·99-s + 3·100-s − 8·103-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 3/4·16-s + 0.447·20-s − 1.66·23-s + 3/5·25-s + 1/6·36-s − 0.657·37-s + 0.603·44-s + 0.298·45-s − 1.16·47-s + 2/7·49-s + 1.64·53-s + 1.07·55-s − 1.04·59-s − 7/8·64-s − 1.95·67-s − 0.670·80-s + 1/9·81-s + 2.11·89-s − 0.834·92-s + 0.406·97-s + 0.402·99-s + 3/10·100-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631279343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631279343\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.43367922372521893051964858193, −10.16697244800954672853871200065, −9.553029277486857738933374806752, −9.042500371866569961928145815452, −8.663929472439463611181209190733, −7.82879275469451888343049520257, −7.24116568379398427399123774874, −6.63888335870128545131813766236, −6.22054270498443237010243029686, −5.72983912823801290762450018660, −4.80705028909179823780348759800, −4.17838209532708222860133644079, −3.36664798148636445321016871164, −2.26934879072045400115587601683, −1.60377190555585889840110248971,
1.60377190555585889840110248971, 2.26934879072045400115587601683, 3.36664798148636445321016871164, 4.17838209532708222860133644079, 4.80705028909179823780348759800, 5.72983912823801290762450018660, 6.22054270498443237010243029686, 6.63888335870128545131813766236, 7.24116568379398427399123774874, 7.82879275469451888343049520257, 8.663929472439463611181209190733, 9.042500371866569961928145815452, 9.553029277486857738933374806752, 10.16697244800954672853871200065, 10.43367922372521893051964858193
