| L(s) = 1 | + 3-s − 5-s − 3·9-s − 11-s − 15-s − 17-s − 19-s − 23-s − 3·25-s − 4·27-s − 4·29-s − 31-s − 33-s + 37-s − 8·41-s + 2·43-s + 3·45-s + 9·47-s − 7·49-s − 51-s − 7·53-s + 55-s − 57-s − 11·59-s − 61-s + 11·67-s − 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 9-s − 0.301·11-s − 0.258·15-s − 0.242·17-s − 0.229·19-s − 0.208·23-s − 3/5·25-s − 0.769·27-s − 0.742·29-s − 0.179·31-s − 0.174·33-s + 0.164·37-s − 1.24·41-s + 0.304·43-s + 0.447·45-s + 1.31·47-s − 49-s − 0.140·51-s − 0.961·53-s + 0.134·55-s − 0.132·57-s − 1.43·59-s − 0.128·61-s + 1.34·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11 T + 112 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 96 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 80 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−14.2994457089, −13.8613988254, −13.5142171080, −12.9767181117, −12.5797917624, −12.0093042393, −11.6580067751, −11.2003237093, −10.7640090979, −10.3666716942, −9.57262362257, −9.34598125753, −8.79180811289, −8.25686320172, −7.97492970204, −7.50456170618, −6.83885242267, −6.27307249415, −5.68158959703, −5.20659420052, −4.44353578872, −3.82597815984, −3.19973068379, −2.59878121834, −1.75620021521, 0,
1.75620021521, 2.59878121834, 3.19973068379, 3.82597815984, 4.44353578872, 5.20659420052, 5.68158959703, 6.27307249415, 6.83885242267, 7.50456170618, 7.97492970204, 8.25686320172, 8.79180811289, 9.34598125753, 9.57262362257, 10.3666716942, 10.7640090979, 11.2003237093, 11.6580067751, 12.0093042393, 12.5797917624, 12.9767181117, 13.5142171080, 13.8613988254, 14.2994457089
