L(s) = 1 | − 3-s − 4-s + 9-s − 11-s + 12-s + 13-s − 3·16-s − 3·19-s + 2·23-s − 4·25-s − 27-s + 5·29-s + 33-s − 36-s + 9·37-s − 39-s − 13·43-s + 44-s + 3·48-s − 2·49-s − 52-s − 21·53-s + 3·57-s − 2·59-s + 61-s + 7·64-s + 15·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 3/4·16-s − 0.688·19-s + 0.417·23-s − 4/5·25-s − 0.192·27-s + 0.928·29-s + 0.174·33-s − 1/6·36-s + 1.47·37-s − 0.160·39-s − 1.98·43-s + 0.150·44-s + 0.433·48-s − 2/7·49-s − 0.138·52-s − 2.88·53-s + 0.397·57-s − 0.260·59-s + 0.128·61-s + 7/8·64-s + 1.83·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100035 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100035 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 106 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 15 T + 186 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 200 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−14.1707939206, −13.8299200307, −13.2488223221, −13.0497342646, −12.6098566004, −12.0897064317, −11.5444984495, −11.1230232219, −10.9652909803, −10.1166848445, −9.88206231076, −9.42485414204, −8.81570795737, −8.31730012303, −7.96181023587, −7.30848075501, −6.60411049064, −6.36175376207, −5.75126577919, −4.96583379025, −4.68790183904, −4.08566126266, −3.29708897293, −2.47161687620, −1.45909861130, 0,
1.45909861130, 2.47161687620, 3.29708897293, 4.08566126266, 4.68790183904, 4.96583379025, 5.75126577919, 6.36175376207, 6.60411049064, 7.30848075501, 7.96181023587, 8.31730012303, 8.81570795737, 9.42485414204, 9.88206231076, 10.1166848445, 10.9652909803, 11.1230232219, 11.5444984495, 12.0897064317, 12.6098566004, 13.0497342646, 13.2488223221, 13.8299200307, 14.1707939206