L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·5-s − 2·6-s − 2·9-s + 4·10-s + 2·12-s − 2·15-s − 4·16-s + 4·18-s − 4·20-s − 25-s − 5·27-s + 4·30-s − 4·31-s + 8·32-s − 4·36-s + 4·45-s − 4·48-s − 49-s + 2·50-s + 12·53-s + 10·54-s − 4·60-s + 8·62-s − 8·64-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s − 2/3·9-s + 1.26·10-s + 0.577·12-s − 0.516·15-s − 16-s + 0.942·18-s − 0.894·20-s − 1/5·25-s − 0.962·27-s + 0.730·30-s − 0.718·31-s + 1.41·32-s − 2/3·36-s + 0.596·45-s − 0.577·48-s − 1/7·49-s + 0.282·50-s + 1.64·53-s + 1.36·54-s − 0.516·60-s + 1.01·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.087603252825030825454636123237, −7.85090293890936197901505599254, −7.37169265619738966914624232776, −7.12687213416235257491801163420, −6.46018095183604781320211722199, −5.97605287256255819865787795231, −5.36990522153821285718601593292, −4.81896397167360608395712600546, −4.17544475261394474161423450299, −3.67951146709553747460997942187, −3.19773560747257360737020214706, −2.37900035147794354357639857446, −1.96935921425456951578292518269, −0.917061660470196146908124227359, 0,
0.917061660470196146908124227359, 1.96935921425456951578292518269, 2.37900035147794354357639857446, 3.19773560747257360737020214706, 3.67951146709553747460997942187, 4.17544475261394474161423450299, 4.81896397167360608395712600546, 5.36990522153821285718601593292, 5.97605287256255819865787795231, 6.46018095183604781320211722199, 7.12687213416235257491801163420, 7.37169265619738966914624232776, 7.85090293890936197901505599254, 8.087603252825030825454636123237