| L(s) = 1 | − 2·2-s + 2·4-s − 4·7-s − 3·9-s − 4·11-s + 8·14-s − 4·16-s + 6·18-s + 8·22-s + 25-s − 8·28-s − 12·29-s + 8·32-s − 6·36-s − 8·44-s + 9·49-s − 2·50-s + 20·53-s + 24·58-s + 12·63-s − 8·64-s + 16·77-s + 4·79-s + 9·81-s − 18·98-s + 12·99-s + 2·100-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.51·7-s − 9-s − 1.20·11-s + 2.13·14-s − 16-s + 1.41·18-s + 1.70·22-s + 1/5·25-s − 1.51·28-s − 2.22·29-s + 1.41·32-s − 36-s − 1.20·44-s + 9/7·49-s − 0.282·50-s + 2.74·53-s + 3.15·58-s + 1.51·63-s − 64-s + 1.82·77-s + 0.450·79-s + 81-s − 1.81·98-s + 1.20·99-s + 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−8.135112042520714114558404572284, −7.74458105352402280406521889140, −7.21774916822495620658805651241, −7.06714769689836528304561655366, −6.35310139484486270129848332695, −5.92117721440141096888049495475, −5.49304995963570350420116364338, −5.05067479105326240090293282252, −4.20722717552085420006375719918, −3.59590050473976856483561667568, −3.09700600809821649267659580968, −2.43389240566778123798750284514, −2.02711293342090032698186573773, −0.71035344490981396455558214488, 0,
0.71035344490981396455558214488, 2.02711293342090032698186573773, 2.43389240566778123798750284514, 3.09700600809821649267659580968, 3.59590050473976856483561667568, 4.20722717552085420006375719918, 5.05067479105326240090293282252, 5.49304995963570350420116364338, 5.92117721440141096888049495475, 6.35310139484486270129848332695, 7.06714769689836528304561655366, 7.21774916822495620658805651241, 7.74458105352402280406521889140, 8.135112042520714114558404572284
