| L(s) = 1 | − 2·5-s − 6·13-s + 8·17-s − 7·25-s + 2·29-s − 16·37-s + 10·41-s − 11·49-s − 16·53-s − 14·61-s + 12·65-s − 24·73-s − 16·85-s − 8·89-s − 6·97-s − 26·101-s − 2·113-s − 19·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.66·13-s + 1.94·17-s − 7/5·25-s + 0.371·29-s − 2.63·37-s + 1.56·41-s − 1.57·49-s − 2.19·53-s − 1.79·61-s + 1.48·65-s − 2.80·73-s − 1.73·85-s − 0.847·89-s − 0.609·97-s − 2.58·101-s − 0.188·113-s − 1.72·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.332·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 131 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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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
−8.397031696841650725913479800939, −8.386977972874860395998300352468, −7.73171250544765767748752303615, −7.68295481383241906673174548290, −7.17607364241454461330902880889, −7.11900592248187017770817754217, −6.18361750459146836249569166071, −6.13060571298140938060085749426, −5.47678506461691529624909531985, −5.18091213208347521217114168871, −4.67401161869370202993917240883, −4.42419700266677387406096061366, −3.63664177750927853708552472010, −3.58986898833385397332573782654, −2.83227874127784074527249501031, −2.65908800261882956258027579143, −1.56974610630703964291935431360, −1.47329622182797242636351288162, 0, 0,
1.47329622182797242636351288162, 1.56974610630703964291935431360, 2.65908800261882956258027579143, 2.83227874127784074527249501031, 3.58986898833385397332573782654, 3.63664177750927853708552472010, 4.42419700266677387406096061366, 4.67401161869370202993917240883, 5.18091213208347521217114168871, 5.47678506461691529624909531985, 6.13060571298140938060085749426, 6.18361750459146836249569166071, 7.11900592248187017770817754217, 7.17607364241454461330902880889, 7.68295481383241906673174548290, 7.73171250544765767748752303615, 8.386977972874860395998300352468, 8.397031696841650725913479800939
