| L(s) = 1 | − 4-s − 9-s + 16-s − 2·19-s − 12·29-s + 4·31-s + 36-s + 12·41-s − 2·49-s + 24·59-s + 28·61-s − 64-s + 2·76-s + 20·79-s + 81-s + 12·89-s + 32·109-s + 12·116-s − 22·121-s − 4·124-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 − 0.458·19-s − 2.22·29-s + 0.718·31-s + 1/6·36-s + 1.87·41-s − 2/7·49-s + 3.12·59-s + 3.58·61-s − 1/8·64-s + 0.229·76-s + 2.25·79-s + 1/9·81-s + 1.27·89-s + 3.06·109-s + 1.11·116-s − 2·121-s − 0.359·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.904614134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904614134\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.762601137633688930135565974823, −8.707924546066177169490290949439, −8.320189834051550421184466012893, −7.84790887721582945729195550272, −7.45388836060171032454165352073, −7.21488981173211428289124788735, −6.69157433898784711523042244955, −6.13319976906138436544438566732, −6.06219415243581732653210482748, −5.36276075021088364313858603462, −5.15144791684935465233929567914, −4.83656939869977604559892487213, −4.01701106225434364186231577418, −3.74572846576651904593028088118, −3.71737891447654124832833402850, −2.75440449176235264017235152792, −2.29966579180626510198145790575, −2.01538314346297996997146711218, −1.00450375836979626562020635740, −0.53151822188718079537270376493,
0.53151822188718079537270376493, 1.00450375836979626562020635740, 2.01538314346297996997146711218, 2.29966579180626510198145790575, 2.75440449176235264017235152792, 3.71737891447654124832833402850, 3.74572846576651904593028088118, 4.01701106225434364186231577418, 4.83656939869977604559892487213, 5.15144791684935465233929567914, 5.36276075021088364313858603462, 6.06219415243581732653210482748, 6.13319976906138436544438566732, 6.69157433898784711523042244955, 7.21488981173211428289124788735, 7.45388836060171032454165352073, 7.84790887721582945729195550272, 8.320189834051550421184466012893, 8.707924546066177169490290949439, 8.762601137633688930135565974823
