| L(s) = 1 | − 4-s + 16-s + 4·17-s + 6·25-s − 12·29-s − 8·43-s − 2·49-s + 20·53-s − 4·61-s − 64-s − 4·68-s + 16·79-s − 6·100-s − 4·101-s − 32·103-s − 24·107-s + 12·113-s + 12·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s + 0.970·17-s + 6/5·25-s − 2.22·29-s − 1.21·43-s − 2/7·49-s + 2.74·53-s − 0.512·61-s − 1/8·64-s − 0.485·68-s + 1.80·79-s − 3/5·100-s − 0.398·101-s − 3.15·103-s − 2.32·107-s + 1.12·113-s + 1.11·116-s + 6/11·121-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 + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726165641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726165641\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.042753150987512427971315816553, −8.399547444869824165629702392029, −8.299197635418708122260273390729, −7.75408384712726716288203695171, −7.49367118316549753137913603084, −6.98489229009012974886029995967, −6.75882412606412976347109517664, −6.27999142065331352449559284997, −5.67352763690676226247897342692, −5.33017944940951941812404765582, −5.31666816703726764753634668059, −4.65809966085322164903435216152, −4.03167131379461858421417338369, −3.93248086319688664586424341691, −3.28568409364300103593474278941, −2.98752763847768671175599656066, −2.31220777880638730092996628525, −1.75061731713770538325996621162, −1.16746380394535865093387629751, −0.45467745625391820476027404625,
0.45467745625391820476027404625, 1.16746380394535865093387629751, 1.75061731713770538325996621162, 2.31220777880638730092996628525, 2.98752763847768671175599656066, 3.28568409364300103593474278941, 3.93248086319688664586424341691, 4.03167131379461858421417338369, 4.65809966085322164903435216152, 5.31666816703726764753634668059, 5.33017944940951941812404765582, 5.67352763690676226247897342692, 6.27999142065331352449559284997, 6.75882412606412976347109517664, 6.98489229009012974886029995967, 7.49367118316549753137913603084, 7.75408384712726716288203695171, 8.299197635418708122260273390729, 8.399547444869824165629702392029, 9.042753150987512427971315816553
