| L(s) = 1 | + 4·9-s − 8·13-s + 6·17-s − 8·19-s + 2·25-s + 16·43-s − 24·47-s − 4·49-s − 12·53-s − 8·67-s + 7·81-s + 24·89-s + 24·101-s + 16·103-s − 32·117-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 2.21·13-s + 1.45·17-s − 1.83·19-s + 2/5·25-s + 2.43·43-s − 3.50·47-s − 4/7·49-s − 1.64·53-s − 0.977·67-s + 7/9·81-s + 2.54·89-s + 2.38·101-s + 1.57·103-s − 2.95·117-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8285555715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8285555715\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−14.78314990195680939291483258019, −14.61407234852838053605671649478, −14.29845803941890501871180999532, −13.12990486843229870070419757162, −12.83958162474705741517128327771, −12.42812820711201065307788952263, −11.96651015706018560929811900846, −11.09976168376602818722871050760, −10.45018406564663794713022036780, −9.859947069031234632947769625138, −9.668885494194043070932761404458, −8.775324792952608305234038274451, −7.73194344000881192909461523211, −7.60550254480635953750503585943, −6.73888792302150141204335430044, −6.06047072166510840925822598058, −4.78095038515256975486634747518, −4.64030908745122342322785253550, −3.31155621017059602303168202463, −2.02390448372635546016211511373,
2.02390448372635546016211511373, 3.31155621017059602303168202463, 4.64030908745122342322785253550, 4.78095038515256975486634747518, 6.06047072166510840925822598058, 6.73888792302150141204335430044, 7.60550254480635953750503585943, 7.73194344000881192909461523211, 8.775324792952608305234038274451, 9.668885494194043070932761404458, 9.859947069031234632947769625138, 10.45018406564663794713022036780, 11.09976168376602818722871050760, 11.96651015706018560929811900846, 12.42812820711201065307788952263, 12.83958162474705741517128327771, 13.12990486843229870070419757162, 14.29845803941890501871180999532, 14.61407234852838053605671649478, 14.78314990195680939291483258019
