| L(s) = 1 | − 7-s − 4·9-s + 6·11-s − 2·23-s + 2·25-s + 6·29-s + 6·37-s + 14·43-s + 49-s + 4·53-s + 4·63-s − 4·67-s + 4·71-s − 6·77-s + 7·81-s − 24·99-s − 4·107-s + 30·109-s − 16·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 4/3·9-s + 1.80·11-s − 0.417·23-s + 2/5·25-s + 1.11·29-s + 0.986·37-s + 2.13·43-s + 1/7·49-s + 0.549·53-s + 0.503·63-s − 0.488·67-s + 0.474·71-s − 0.683·77-s + 7/9·81-s − 2.41·99-s − 0.386·107-s + 2.87·109-s − 1.50·113-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.003595771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.003595771\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.971546368458368402477222584228, −7.52524187274780758002045715058, −6.98524175315750446329803899079, −6.58417639129110548206174139222, −6.13891864417063180016649951747, −5.93714354268193916935828673496, −5.42294687986564573940240116382, −4.75753626262456766060320377778, −4.27065933609661481678911430733, −3.86148798444796531892510686273, −3.32222335300377067084853089645, −2.71893794562589817047343582243, −2.32529761773487740473529909758, −1.33728408232782050400274817280, −0.66605993241860822215856807506,
0.66605993241860822215856807506, 1.33728408232782050400274817280, 2.32529761773487740473529909758, 2.71893794562589817047343582243, 3.32222335300377067084853089645, 3.86148798444796531892510686273, 4.27065933609661481678911430733, 4.75753626262456766060320377778, 5.42294687986564573940240116382, 5.93714354268193916935828673496, 6.13891864417063180016649951747, 6.58417639129110548206174139222, 6.98524175315750446329803899079, 7.52524187274780758002045715058, 7.971546368458368402477222584228
