| L(s) = 1 | − 2·4-s − 3·7-s + 4·9-s − 2·11-s + 3·13-s − 3·17-s + 2·25-s + 6·28-s + 5·29-s − 8·36-s − 18·37-s + 2·43-s + 4·44-s − 47-s + 2·49-s − 6·52-s − 9·53-s + 3·59-s − 12·63-s + 8·64-s + 6·68-s + 6·77-s + 7·81-s − 89-s − 9·91-s + 7·97-s − 8·99-s + ⋯ |
| L(s) = 1 | − 4-s − 1.13·7-s + 4/3·9-s − 0.603·11-s + 0.832·13-s − 0.727·17-s + 2/5·25-s + 1.13·28-s + 0.928·29-s − 4/3·36-s − 2.95·37-s + 0.304·43-s + 0.603·44-s − 0.145·47-s + 2/7·49-s − 0.832·52-s − 1.23·53-s + 0.390·59-s − 1.51·63-s + 64-s + 0.727·68-s + 0.683·77-s + 7/9·81-s − 0.105·89-s − 0.943·91-s + 0.710·97-s − 0.804·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 53 | $C_2$ | \( 1 + 9 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 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$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 123 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.050101228573636708266372025897, −8.766018936495003298267649534352, −8.279822783231006762782106943520, −7.66248074781825646191060287299, −6.91886529036198885434978535395, −6.76597116571120970985488675933, −6.22215952522432837442809778303, −5.44338505733965197411278194218, −4.86124192771639196452864079302, −4.48268419403888396874177731970, −3.72872072357510651370465562067, −3.39109348630135161261968544752, −2.41764485281868085168943670532, −1.36692709868057597547828932752, 0,
1.36692709868057597547828932752, 2.41764485281868085168943670532, 3.39109348630135161261968544752, 3.72872072357510651370465562067, 4.48268419403888396874177731970, 4.86124192771639196452864079302, 5.44338505733965197411278194218, 6.22215952522432837442809778303, 6.76597116571120970985488675933, 6.91886529036198885434978535395, 7.66248074781825646191060287299, 8.279822783231006762782106943520, 8.766018936495003298267649534352, 9.050101228573636708266372025897
