| L(s) = 1 | − 4-s − 9-s + 16-s − 8·19-s − 4·29-s + 36-s − 8·41-s + 10·49-s + 4·59-s − 28·61-s − 64-s − 12·71-s + 8·76-s + 24·79-s + 81-s + 20·89-s − 8·101-s − 12·109-s + 4·116-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s − 1.83·19-s − 0.742·29-s + 1/6·36-s − 1.24·41-s + 10/7·49-s + 0.520·59-s − 3.58·61-s − 1/8·64-s − 1.42·71-s + 0.917·76-s + 2.70·79-s + 1/9·81-s + 2.11·89-s − 0.796·101-s − 1.14·109-s + 0.371·116-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7413590772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7413590772\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.110788208929085661582988145167, −8.726612759448717761676608627680, −8.245713425672009827702196475572, −8.171645687220001427327486960866, −7.51253376315177692496404071230, −7.32700993895320354554923406000, −6.66347583389724343324811339243, −6.43136247515549689745410535311, −5.97354125526828108453193730752, −5.65613112164450728550952185048, −5.10887514205157063566702842108, −4.76779435809111368358280410886, −4.17811415120883478134440091476, −4.10377080365888073746864053291, −3.23468809516091379113423010245, −3.17237105256240963754999452597, −2.19495990950516027468695589797, −2.03246840653963387152554899326, −1.22127075589787683742109683507, −0.29829319085068413111911254106,
0.29829319085068413111911254106, 1.22127075589787683742109683507, 2.03246840653963387152554899326, 2.19495990950516027468695589797, 3.17237105256240963754999452597, 3.23468809516091379113423010245, 4.10377080365888073746864053291, 4.17811415120883478134440091476, 4.76779435809111368358280410886, 5.10887514205157063566702842108, 5.65613112164450728550952185048, 5.97354125526828108453193730752, 6.43136247515549689745410535311, 6.66347583389724343324811339243, 7.32700993895320354554923406000, 7.51253376315177692496404071230, 8.171645687220001427327486960866, 8.245713425672009827702196475572, 8.726612759448717761676608627680, 9.110788208929085661582988145167
