L(s) = 1 | − 3·2-s + 2·3-s + 4·4-s − 6·6-s − 3·8-s + 3·9-s − 2·11-s + 8·12-s + 2·13-s + 3·16-s − 12·17-s − 9·18-s + 4·19-s + 6·22-s + 4·23-s − 6·24-s − 5·25-s − 6·26-s + 4·27-s − 16·29-s − 8·31-s − 6·32-s − 4·33-s + 36·34-s + 12·36-s − 10·37-s − 12·38-s + ⋯ |
L(s) = 1 | − 2.12·2-s + 1.15·3-s + 2·4-s − 2.44·6-s − 1.06·8-s + 9-s − 0.603·11-s + 2.30·12-s + 0.554·13-s + 3/4·16-s − 2.91·17-s − 2.12·18-s + 0.917·19-s + 1.27·22-s + 0.834·23-s − 1.22·24-s − 25-s − 1.17·26-s + 0.769·27-s − 2.97·29-s − 1.43·31-s − 1.06·32-s − 0.696·33-s + 6.17·34-s + 2·36-s − 1.64·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 16 T + 117 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.136129125392656469050322013341, −8.807030341900377071986106821698, −8.458543213076420806943974649736, −8.410384676094100896686091956837, −7.60782286100225971478383860507, −7.52275297217827678920123400387, −7.03252398659051344961708471276, −6.84612894136881739739190128481, −6.08458139083664908339960167012, −5.62692514355561985503532827542, −5.08833377946787577344317188999, −4.54919738362808015152712971308, −3.92670638612145572212418942778, −3.40230264114730994288873085529, −3.12806557848555480053285195221, −2.16099573002650604749150685811, −1.84069596447255548673900203893, −1.51088916595908606697405159221, 0, 0,
1.51088916595908606697405159221, 1.84069596447255548673900203893, 2.16099573002650604749150685811, 3.12806557848555480053285195221, 3.40230264114730994288873085529, 3.92670638612145572212418942778, 4.54919738362808015152712971308, 5.08833377946787577344317188999, 5.62692514355561985503532827542, 6.08458139083664908339960167012, 6.84612894136881739739190128481, 7.03252398659051344961708471276, 7.52275297217827678920123400387, 7.60782286100225971478383860507, 8.410384676094100896686091956837, 8.458543213076420806943974649736, 8.807030341900377071986106821698, 9.136129125392656469050322013341