| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 4·5-s + 8·6-s − 4·7-s + 6·9-s + 8·10-s − 8·12-s − 6·13-s + 8·14-s + 16·15-s − 4·16-s − 6·17-s − 12·18-s − 8·20-s + 16·21-s − 4·23-s + 6·25-s + 12·26-s + 4·27-s − 8·28-s − 32·30-s + 4·31-s + 8·32-s + 12·34-s + 16·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 1.78·5-s + 3.26·6-s − 1.51·7-s + 2·9-s + 2.52·10-s − 2.30·12-s − 1.66·13-s + 2.13·14-s + 4.13·15-s − 16-s − 1.45·17-s − 2.82·18-s − 1.78·20-s + 3.49·21-s − 0.834·23-s + 6/5·25-s + 2.35·26-s + 0.769·27-s − 1.51·28-s − 5.84·30-s + 0.718·31-s + 1.41·32-s + 2.05·34-s + 2.70·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−17.4780977310, −16.8238626428, −16.8215772884, −15.9802273957, −15.9783584270, −15.5519122270, −14.7879696813, −13.9309448252, −13.0237416618, −12.3834323879, −12.2448940378, −11.5135334923, −11.3244064839, −10.7971801876, −10.2584581032, −9.56682706021, −9.20679535768, −8.01633913391, −7.86442320164, −6.82871744579, −6.60597695324, −5.97057694071, −4.69108797829, −4.49472027398, −2.93953492658, 0, 0,
2.93953492658, 4.49472027398, 4.69108797829, 5.97057694071, 6.60597695324, 6.82871744579, 7.86442320164, 8.01633913391, 9.20679535768, 9.56682706021, 10.2584581032, 10.7971801876, 11.3244064839, 11.5135334923, 12.2448940378, 12.3834323879, 13.0237416618, 13.9309448252, 14.7879696813, 15.5519122270, 15.9783584270, 15.9802273957, 16.8215772884, 16.8238626428, 17.4780977310
