| L(s) = 1 | + 4-s − 2·7-s + 4·13-s + 16-s − 8·19-s + 25-s − 2·28-s − 8·31-s + 16·37-s + 16·43-s − 11·49-s + 4·52-s − 26·61-s + 64-s − 26·67-s − 8·73-s − 8·76-s − 20·79-s − 8·91-s + 4·97-s + 100-s + 16·103-s − 14·109-s − 2·112-s + 14·121-s − 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s + 1.10·13-s + 1/4·16-s − 1.83·19-s + 1/5·25-s − 0.377·28-s − 1.43·31-s + 2.63·37-s + 2.43·43-s − 1.57·49-s + 0.554·52-s − 3.32·61-s + 1/8·64-s − 3.17·67-s − 0.936·73-s − 0.917·76-s − 2.25·79-s − 0.838·91-s + 0.406·97-s + 1/10·100-s + 1.57·103-s − 1.34·109-s − 0.188·112-s + 1.27·121-s − 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.042506553318738798296336940319, −7.55608614338171334988141719926, −7.42407483081609479054268881777, −6.62750192113583511421649895960, −6.20126311780820576334465009943, −5.99605261483170121408204274618, −5.73490641215111323195434065331, −4.63365408594636780088580922329, −4.34374909165545672967941970382, −3.90981816402000462440548769015, −2.95626008175642733558542036272, −2.92721937122489965596579237904, −1.94341038324491950956024540703, −1.29915321561212133045480179839, 0,
1.29915321561212133045480179839, 1.94341038324491950956024540703, 2.92721937122489965596579237904, 2.95626008175642733558542036272, 3.90981816402000462440548769015, 4.34374909165545672967941970382, 4.63365408594636780088580922329, 5.73490641215111323195434065331, 5.99605261483170121408204274618, 6.20126311780820576334465009943, 6.62750192113583511421649895960, 7.42407483081609479054268881777, 7.55608614338171334988141719926, 8.042506553318738798296336940319
