| L(s) = 1 | + 4·3-s − 4-s + 6·9-s − 4·12-s + 4·13-s + 16-s − 12·17-s − 25-s − 4·27-s + 12·29-s − 6·36-s + 16·39-s − 20·43-s + 4·48-s + 14·49-s − 48·51-s − 4·52-s + 20·61-s − 64-s + 12·68-s − 4·75-s − 16·79-s − 37·81-s + 48·87-s + 100-s + 12·101-s − 8·103-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1/2·4-s + 2·9-s − 1.15·12-s + 1.10·13-s + 1/4·16-s − 2.91·17-s − 1/5·25-s − 0.769·27-s + 2.22·29-s − 36-s + 2.56·39-s − 3.04·43-s + 0.577·48-s + 2·49-s − 6.72·51-s − 0.554·52-s + 2.56·61-s − 1/8·64-s + 1.45·68-s − 0.461·75-s − 1.80·79-s − 4.11·81-s + 5.14·87-s + 1/10·100-s + 1.19·101-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.881863092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881863092\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−13.64307047223434482753417733184, −13.43987587270077914606363645205, −12.97672677219211980506742795396, −12.15278791750272641970502876374, −11.34035261918977020073503627788, −11.16198642625201982590408812968, −10.12234744829263090851293025623, −9.892917837024798160963779878530, −8.925613416825172513155250554034, −8.798807137075199390827783279547, −8.406467371345004101137487683045, −8.225449997921241172760293684223, −7.12850970394112456487857799337, −6.69135004082542710953257671703, −5.87667466683688625617498021739, −4.77795400527083453366709640303, −4.12831713785439225613911682394, −3.49913214415705381439925680590, −2.70495821028726688122985006903, −2.02481506755937410809719002151,
2.02481506755937410809719002151, 2.70495821028726688122985006903, 3.49913214415705381439925680590, 4.12831713785439225613911682394, 4.77795400527083453366709640303, 5.87667466683688625617498021739, 6.69135004082542710953257671703, 7.12850970394112456487857799337, 8.225449997921241172760293684223, 8.406467371345004101137487683045, 8.798807137075199390827783279547, 8.925613416825172513155250554034, 9.892917837024798160963779878530, 10.12234744829263090851293025623, 11.16198642625201982590408812968, 11.34035261918977020073503627788, 12.15278791750272641970502876374, 12.97672677219211980506742795396, 13.43987587270077914606363645205, 13.64307047223434482753417733184
