L(s) = 1 | − 2-s + 3·3-s + 2·5-s − 3·6-s + 8-s + 6·9-s − 2·10-s − 4·11-s − 2·13-s + 6·15-s − 16-s − 6·18-s + 7·19-s + 4·22-s + 6·23-s + 3·24-s − 7·25-s + 2·26-s + 9·27-s + 8·29-s − 6·30-s − 4·31-s − 12·33-s + 6·37-s − 7·38-s − 6·39-s + 2·40-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 0.894·5-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.632·10-s − 1.20·11-s − 0.554·13-s + 1.54·15-s − 1/4·16-s − 1.41·18-s + 1.60·19-s + 0.852·22-s + 1.25·23-s + 0.612·24-s − 7/5·25-s + 0.392·26-s + 1.73·27-s + 1.48·29-s − 1.09·30-s − 0.718·31-s − 2.08·33-s + 0.986·37-s − 1.13·38-s − 0.960·39-s + 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.089441585\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.089441585\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.16120251755305790315214905013, −9.550027393577338302587892936634, −9.418451306269303266436337050564, −9.362265645578935274845686490731, −8.804571992395202177784237954641, −8.175946341675684565961940566617, −7.80647970034360018454113821016, −7.62556891049431017378237188103, −7.31121778961765575346969382793, −6.67859669866395251915256785254, −6.01684802293114657038083994860, −5.44693688636994928786215129611, −5.17918794070119069559104106739, −4.25973770316633115601407576025, −4.16283524735691565479251443880, −2.95036244465522895124321159919, −2.94785854990316999372241917477, −2.38238348871946973393256561733, −1.68369557891561626232110444018, −0.905410489371427862173649299393,
0.905410489371427862173649299393, 1.68369557891561626232110444018, 2.38238348871946973393256561733, 2.94785854990316999372241917477, 2.95036244465522895124321159919, 4.16283524735691565479251443880, 4.25973770316633115601407576025, 5.17918794070119069559104106739, 5.44693688636994928786215129611, 6.01684802293114657038083994860, 6.67859669866395251915256785254, 7.31121778961765575346969382793, 7.62556891049431017378237188103, 7.80647970034360018454113821016, 8.175946341675684565961940566617, 8.804571992395202177784237954641, 9.362265645578935274845686490731, 9.418451306269303266436337050564, 9.550027393577338302587892936634, 10.16120251755305790315214905013