| L(s) = 1 | + 4-s − 2·9-s + 16-s + 4·19-s − 5·25-s − 12·29-s − 8·31-s − 2·36-s + 12·41-s + 49-s − 12·59-s + 16·61-s + 64-s + 4·76-s + 16·79-s − 5·81-s − 12·89-s − 5·100-s + 4·109-s − 12·116-s − 22·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/3·9-s + 1/4·16-s + 0.917·19-s − 25-s − 2.22·29-s − 1.43·31-s − 1/3·36-s + 1.87·41-s + 1/7·49-s − 1.56·59-s + 2.04·61-s + 1/8·64-s + 0.458·76-s + 1.80·79-s − 5/9·81-s − 1.27·89-s − 1/2·100-s + 0.383·109-s − 1.11·116-s − 2·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8778167717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8778167717\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−12.30526099005271094386573698884, −11.46841286601878794911833201781, −11.23136141438460806904855801814, −10.74873683533699837976941407431, −9.765547119459919407856461234632, −9.395997320196623402140164716317, −8.758164081717971221129539824522, −7.71895664384636679391588711627, −7.57571100088867902110310233811, −6.66965130292041403123447657278, −5.59857578974307577358363282055, −5.57928681742950427486583645839, −4.11500666640753372808858013263, −3.28327332646103144207952017478, −2.07204529193989626181377807359,
2.07204529193989626181377807359, 3.28327332646103144207952017478, 4.11500666640753372808858013263, 5.57928681742950427486583645839, 5.59857578974307577358363282055, 6.66965130292041403123447657278, 7.57571100088867902110310233811, 7.71895664384636679391588711627, 8.758164081717971221129539824522, 9.395997320196623402140164716317, 9.765547119459919407856461234632, 10.74873683533699837976941407431, 11.23136141438460806904855801814, 11.46841286601878794911833201781, 12.30526099005271094386573698884
