| L(s) = 1 | − 2·3-s − 2·5-s − 4·7-s + 3·9-s + 2·11-s + 4·13-s + 4·15-s − 4·19-s + 8·21-s + 3·25-s − 4·27-s + 8·31-s − 4·33-s + 8·35-s + 4·37-s − 8·39-s − 4·43-s − 6·45-s − 2·49-s − 12·53-s − 4·55-s + 8·57-s + 4·61-s − 12·63-s − 8·65-s − 16·67-s + 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1.51·7-s + 9-s + 0.603·11-s + 1.10·13-s + 1.03·15-s − 0.917·19-s + 1.74·21-s + 3/5·25-s − 0.769·27-s + 1.43·31-s − 0.696·33-s + 1.35·35-s + 0.657·37-s − 1.28·39-s − 0.609·43-s − 0.894·45-s − 2/7·49-s − 1.64·53-s − 0.539·55-s + 1.05·57-s + 0.512·61-s − 1.51·63-s − 0.992·65-s − 1.95·67-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6969600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6969600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.496569994323460503327956488795, −8.427790968356949828463766069928, −7.76813913138079488830080052517, −7.66211342638569411621269128690, −6.75113200070357754517658658712, −6.68271084153030993878430940072, −6.40787268575078923072790599149, −6.26154275359439917132319686044, −5.58304828298920242880885066352, −5.25084257901093312563813702669, −4.65875871392749080481063307709, −4.19323252670282849197751823125, −3.94757883397237131132630419607, −3.57677512440858911860405383850, −2.82873950610865792156420289681, −2.70676311770179879043362401908, −1.34268489597031229488255541043, −1.31312034502833414605901593200, 0, 0,
1.31312034502833414605901593200, 1.34268489597031229488255541043, 2.70676311770179879043362401908, 2.82873950610865792156420289681, 3.57677512440858911860405383850, 3.94757883397237131132630419607, 4.19323252670282849197751823125, 4.65875871392749080481063307709, 5.25084257901093312563813702669, 5.58304828298920242880885066352, 6.26154275359439917132319686044, 6.40787268575078923072790599149, 6.68271084153030993878430940072, 6.75113200070357754517658658712, 7.66211342638569411621269128690, 7.76813913138079488830080052517, 8.427790968356949828463766069928, 8.496569994323460503327956488795
