| L(s) = 1 | − 4-s − 5-s − 2·9-s − 11-s − 3·16-s + 4·19-s + 20-s + 25-s + 4·31-s + 2·36-s + 44-s + 2·45-s + 14·49-s + 55-s − 12·59-s − 8·61-s + 7·64-s − 4·76-s + 4·79-s + 3·80-s − 5·81-s − 12·89-s − 4·95-s + 2·99-s − 100-s − 12·101-s + 4·109-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 3/4·16-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.718·31-s + 1/3·36-s + 0.150·44-s + 0.298·45-s + 2·49-s + 0.134·55-s − 1.56·59-s − 1.02·61-s + 7/8·64-s − 0.458·76-s + 0.450·79-s + 0.335·80-s − 5/9·81-s − 1.27·89-s − 0.410·95-s + 0.201·99-s − 0.0999·100-s − 1.19·101-s + 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4936014653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4936014653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
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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
−13.75539545875999856307279939656, −13.40790056481005511778602564463, −12.42556471409184263431561392223, −11.99318278851470879678465377104, −11.27748233626068839689230673523, −10.71705090685439114964224517084, −9.850315198815836374223293791035, −9.142886062799315513511972371368, −8.558267557755620760586313944365, −7.78942950669997960453513423290, −7.04063543577616078388107278171, −5.99080016242421450165909429669, −5.09698455816962605625731143596, −4.17584295292025542328208055314, −2.89395902106255541986149724624,
2.89395902106255541986149724624, 4.17584295292025542328208055314, 5.09698455816962605625731143596, 5.99080016242421450165909429669, 7.04063543577616078388107278171, 7.78942950669997960453513423290, 8.558267557755620760586313944365, 9.142886062799315513511972371368, 9.850315198815836374223293791035, 10.71705090685439114964224517084, 11.27748233626068839689230673523, 11.99318278851470879678465377104, 12.42556471409184263431561392223, 13.40790056481005511778602564463, 13.75539545875999856307279939656
