| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 4·8-s + 3·9-s + 4·10-s + 2·11-s + 6·12-s + 2·13-s − 4·15-s + 5·16-s − 2·17-s − 6·18-s − 2·19-s − 6·20-s − 4·22-s − 6·23-s − 8·24-s − 7·25-s − 4·26-s + 4·27-s − 6·29-s + 8·30-s − 6·32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s + 0.603·11-s + 1.73·12-s + 0.554·13-s − 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 0.458·19-s − 1.34·20-s − 0.852·22-s − 1.25·23-s − 1.63·24-s − 7/5·25-s − 0.784·26-s + 0.769·27-s − 1.11·29-s + 1.46·30-s − 1.06·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 169 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 132 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 96 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 3 p T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 176 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 180 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.333450475069854668767617808663, −8.075226220005108109741929851352, −7.67497794974218508017453049677, −7.59117639499942607057924500709, −6.93655044771923842841090375216, −6.60103553890156028753093974428, −6.33428776092417073425145528071, −6.04055493258928954296714878262, −5.14255167184064257574334267988, −5.05641886092505991118724275315, −4.05718100750002673432179103616, −4.00666036152517151685042112178, −3.49272227002787046183537912527, −3.37868070081080166625747246235, −2.37965248857818483110070168218, −2.29050374623032823957853023007, −1.48791601824780027950030084283, −1.44243530125595939482369013939, 0, 0,
1.44243530125595939482369013939, 1.48791601824780027950030084283, 2.29050374623032823957853023007, 2.37965248857818483110070168218, 3.37868070081080166625747246235, 3.49272227002787046183537912527, 4.00666036152517151685042112178, 4.05718100750002673432179103616, 5.05641886092505991118724275315, 5.14255167184064257574334267988, 6.04055493258928954296714878262, 6.33428776092417073425145528071, 6.60103553890156028753093974428, 6.93655044771923842841090375216, 7.59117639499942607057924500709, 7.67497794974218508017453049677, 8.075226220005108109741929851352, 8.333450475069854668767617808663
